c i / S

P201379 N9108 AGARD-AG-192

o»

< •

Q CC

< <

AGARDograph No. 192

Short Course on

Testing of Precision Inertial Gyroscopes

by D.A.Lorenzini

DISTRIBUTION AND AVAILABILITY O N BACK COVER

AGARD-AG-192

NORTH ATLANTIC TREATY ORGANIZATION

ADVISORY GROUP FOR AEROSPACE RESEARCH AND DEVELOPMENT

(ORGANISATION DU TRAITE DE L'ATLANTIQUE NORD)

AGARDograph No. 192

TESTING OF PRECISION INERTIAL GYROSCOPES

by

Dino A.Lorenzini

Major, USAF Chief, Inertial Guidance Research Division

F.J.Seiler Research Laboratory USAF Academy, Colorado 80840, USA

Short Course

April 1973

Italy, Netherlands, Germany, Belgium, United Kingdom, Norway

This AGARDograph was prepared at the request of the Guidance and Control Panel of AGARD-NATO.

THE MISSION OF AGARD

The mission of AGARD is to bring together the leading personalities of the NATO nations in the fields of science and technology relating to aerospace for the following purposes:

- Exchanging of scientific and technical information;

Continuously stimulating advances in the aerospace sciences relevant to strengthening the common defence posture;

- Improving the co-operation among member nations in aerospace research and development;

- Providing scientific and technical advice and assistance to the North Atlantic Military Committee in the field of aerospace research and development;

- Rendering scientific and technical assistance, as requested, to other NATO bodies and to member nations in connection with research and development problems in the aerospace field;

Providing assistance to member nations for the purpose of increasing their scientific and technical potential;

- Recommending effective ways for the member nations to use their research and development capabilities for the common benefit of the NATO community.

The highest authority within AGARD is the National Delegates Board consisting of officially appointed senior representatives from each member nation. The mission of AGARD is carried out through the Panels which are composed of experts appointed by the National Delegates, the Consultant and Exchange Program and the Aerospace Applications Studies Program. The results of AGARD work are reported to the member nations and the NATO Authorities through the AGARD series of publications of which this is one.

Participation in AGARD activities is by invitation only and is normally limited to citizens of the NATO nations.

Published June 1974

681.082.16 : 621.001.4

to Set and printed by Technical Editing and Reproduction Ltd

Harford House. 7-9 Charlotte St. London. W1P HID

PREFACE

This report deals with recent advances which have been made in the testing of precision inertial gyroscopes. The basic phases which are involved in any inertial sensor test are reviewed. These are environment, excitation, monitor and evaluation. Those factors which must be considered in each of these phases to successfully conduct precise gyro tests are discussed. Many new techniques which have been considered or have already been applied to known gyro testing problems are described. The base motion environment, computer-controlled test operation, data acquisition, and data handling problems are emphasized.

Some specialized gyro testing techniques are described in order to provide a sample of current thinking in the areas of tumble testing, error modeling, centrifuge and linear vibration testing. These studies suggest that the development of a more comprehensive gyro error model will uncover some of the coefficient disagreements and instabilities which have been observed from different tests and different test equipment. It is also pointed out that a mini-computer can be effectively used to develop advanced data acquisition and processing methods which will make considerable improvements in terms of test accuracy, speed and versatility.

The conclusion to be reached in this report is that gyro testing has arrived at a new plateau of its evolutionary processes. The challenge which has been presented to the inertial guidance test community by the development of a new generation of highly accurate inertial sensors, demands that some highly innovative thinking be applied to the testing problem if we are to be successful in keeping up with the pace of instrument design and construction. The concepts which are presented are intended to stimulate new ideas in gyro testing and are not to be interpreted as proven techniques or even the most desirable approach to be used.

Dino A.Lorenzini Major, USAF

CONTENTS

Page

iii

vi

vii

viii

I I I 2

2 2 3 3 4 8

13 13 14 16 17 17 17 18 19 21 22 22 23 24

GYRO TESTING TECHNIQUES 26 3.1 Two-Axis Tumble Test 26

3.1.1 Introduction 26 3.1.2 Gyroscope Performance Model 26 3.1.3 Test Description 27 3.1.4 Solution for Drift Coefficients 27 3.1.5 Summary 30

3.2 Gyro Error Modeling 31 3.2.1 Introduction 31 3.2.2 Float Motion Errors 31 3.2.3 Temperature Effects 32 3.2.4 Input Power Variations 32 3.2.5 Test Methods and Analysis Procedure 32 3.2.6 Experimental Results 33 3.2.7 Summary and Conclusions 37

3.3 Precision Centrifuge Test 38 3.3.1 Introduction 38 3.3.2 Test Procedures 38 3.3.3 The Gyro Performance Model 39 3.3.4 A Solution for the Drift Coefficients 42 3.3.5 Test Error Analysis 45 3.3.6 Summary 46

3.4 Vibration Testing 47 3.4.1 Introduction 47 3.4.2 Linear Vibration Versus Tumble Test Compliance Coefficients 49 3.4.3 New Concepts for Gyro Vibration Testing S3 3.4.4 Conclusions 58

PREFACE

LIST OF FIGURES

LIST OF TABLES

NOTATION

1. INTRODUCTION 1.1 1.2 1.3

Background Purpose Scope

2. BASIC TEST CONSIDERATIONS 2.1 2.2

2.3

2.4

2.5

Overview Environment 2.2.1 Definitions 2.2.2 Effects 2.2.3 Techniques Excitation 2.3.1 Test Table 2.3.2 Gyro Electronics 2.3.3 Seismic Simulation Monitor 2.4.1 Background 2.4.2 Data Acquisition 2.4.3 Filtering 2.4.4 Data Handling 2.4.5 Displays Evaluation 2.5.1 Overview 2.5.2 Computer Hardware 2.5.3 Stage

Page

4. CONCLUSIONS 58 4.1 Summary 58 4.2 Future Trends 59 4.3 Recommendations 59

REFERENCES 60

LIST OF FIGURES

Figure 2.1

Figure 2.2(a)

Figure 2.2(b)

Figure 2.3

Figure 2.4

Figure 2.5

Figure 2.6

Figure 2.7

Figure 2.8

Figure 2.9

Figure 2.10

Figure 2.11

Figure 2.12

Figure 2.13

Figure 3.1

Figure 3.2

Figure 3.3

Figure 3.4

Figure 3.5

Figure 3.6

Figure 3.7

Figure 3.8

Figure 3.9

Figure 3.10

Figure 3.11

Figure 3.12

Figure 3.13

Page

Computer-controlled gyro test station block diagram 3

Maximum error in gyroscope testing due to 0.001 arc second of angular vibration 6

Requirements for test platform stability for limiting drift rate error to 0.001 menu

Isolation to base motion inputs 10

Isolation to disturbance forces 11

Orientation of seismometers 12

Tilt sensor control loop 13

Seismometer controlled damping loop 14

Horizontal damping loops 15

Gyro excitation electronics console 16

Transform of signal whose frequency content is less than 1/(2A) 19

Transform of signal whose frequency content is greater than l/(2A) 19

Data processing sequence 20

Flow diagram of test operations software 25

Axes of a single-degree-of-freedom gyro 27

Alignment of gyro and test table for the two-axis tumble test 28

Parameter variation test, spin axis up 34

Tumble test drift rate 35

Residuals from tumble test using classical error model 36

Residuals from tumble test using extended error model 36

Probability density function test residuals using extended error model 37

Gyro orientation for centrifuge test 39

Representation of float and case reference points 50

Float orientation at start of OA//EA North 52

Locus of float center for OA//EA North 52

Gyro orientation for linear vibration test 55

Multiple-input, single-output gyro performance model relationship 57

VI

LIST OF TABLES

Page

Table 3.1 Parameter Variation Test Results 33

Table 3.2 Comparison of Regression Analysis Results for Continuous Tumble Test 35

Table 3.3 Comparison of the Relative Magnitudes of the Gyro Error Model Terms in a Laboratory Test Environment 38

Vl l

NOTATION

A orientation of gyro about its input axis

Aj ith harmonic of sine Fourier coefficient

Ajj sensitivity of gyro to the product of float rotation about the i and j axes

A: sensitivity of gyro drift to angular accelerations about the gyro j axis

A;; term on the jth row of the jth column of the inverse least squares matrix

a linear acceleration vector

av specific force due to linear vibration

B float center of buoyancy

Bj ith harmonic of cosine Fourier coefficient

Bj sensitivity of gyro to the rate of change of temperature along the j axis

D vector of drift coefficients

D vector of generalized float displacements

D calculated value for gyro drift coefficient

Dp fixed gyro drift rate

Dj sensitivity of gyro drift rate to specific forces along the gyro j axis

Djk sensitivity of gyro drift rate to the product of specific forces along the gyro j and

k axes

d separation distance between seismometers

d* displacement of the float along the IA

dfcj displacement of the k end of the float along the j axis

E, drift rate error due to alignment uncertainty with respect to earth rate

E2 drift rate error due to alignment uncertainty with respect to gravity

E3 drift rate error due to rate uncertainty about the gyro input axis

e" vector of residual errors

F vector of generalized float forces

F(f) ideal low-pass filter frequency response

f specific force vector

f specific forces along the gyro j axis

G float geometric center

G displacement of S from B

Gy sensitivity of gyro to the product of temperature gradient along the i axis and specific force along the j axis

G x x ( f ) power spectral density of x

Gx y(f) cross-power spectral density of x and y

g~ gravity vector

gs distance from B to S along the SA

H horizontal seismometer output

H(f) system frequency response function

h(r) system impulse response function

i torque-to-balance current

i(t) impulse function

K matrix of float suspension stiffnesses

K| magnetic suspension stiffness along the IA

L local astronomic latitude angle

M float center of mass

MQ torque due to magnetic suspension forces

N magnetic suspension null point

N number of observations

Ny sensitivity of gyro to the product of float displacements along the i and j axes

P\V i PMS* ^SG • PTG , PH sensitivity of gyro to power variations of the wheel, microsyn supply, signal generator,

torque generator, and heaters

Pj sensitivity of gyro to float displacements along the j axis

q number of significant terms in the error model

R case geometric center

R multiple correlation coefficient

Rj rotation matrix of angle j

Rj sensitivity of gyro drift to angular rates about the gyro j axis

Rj sensitivity of gyro to float rotation about the j axis

Rjk sensitivity of gyro drift to the product of angular rates about the gyro j and k axes

R X X ( T ) auto-correlation function of x

R x y ( r ) cross-correlation function of x and y

rk radius of gyro displacement at the k end of the float

S float center of magnetic suspension force

S J G torque generator scale factor

Sj(f) transform of sampled signal

Sj sensitivity of gyro drift to displacement of the SG end of the float along the j axis

S(t) analog signal which is a function of time

ix

T Q , T R sensitivity of gyro to average gyro (room) temperature

Tj sensitivity of gyro drift to displacements of the TG end of the float along the j axis

t time

. t-test ratio

U displacement of M from B

V vertical seismometer output

Vj sensitivity of gyro to float velocity along the j axis

Wp centrifuge angular rate

Wi£ angular velocity of the earth relative to inertial space

Wu total drift rate of gyro output

Wj sensitivity of gyro to float angular velocity about the j axis

Wp rate table angular rate

w angular frequency (rad/sec)

w: angular acceleration about the platform j axis

X matrix of coordinate functions

x linear acceleration along the platform x axis

Y vector of gyro output observations

7 orientation of gyro about its input axis

72(f) coherency spectrum of system transfer function

6w uncertainty in the gyro output signal

0 gyro misalignment angle

6 rotation angle of centrifuge main arm

0 x , 0 y tilt about platform x(y) axis

a standard error of test

O: standard error of the jth coefficient

T filter time constant

<t> rotation of the test table about minus earth's axis

w j , WQ . " S angular rate about gyro case axes

A time interval between data samples

<J>(f) phase angle of system transfer function

TESTING OF PRECISION INERTIAL GYROSCOPES

Dino A.Lorenzini

1. INTRODUCTION

1.1 Background

The testing of precision inertial guidance components has undergone revolutionary changes in recent years. These changes have been stimulated by the introduction of increasingly more precise instruments. The recent emergence of the Third Generation Gyroscope (TGG) and the Specific Force Integrating Receiver (SFIR) from the C.S. Draper Laboratory promises to improve inertial sensor performance by an order of magnitude. Consequently, a new generation of laboratory test equipment, procedures and data analysis techniques commensurate with the performance goals of these instruments had to be developed.

The stated performance goal for the TGG is 0.001 meru of drift uncertainty1. This amounts to a total drift angle of about one second of arc in twenty-four hours. Extremely small deviations in the orientation of the test specimen with respect to the earth rate vector, and angular rates as small as 0.001 arcseconds per minute will significantly alter the measured performance of such an instrument. In addition, small temperature variations or minute changes in excitation levels during a test will generate erroneous output signals. These numerous disturbing parameters must be properly monitored, modeled, and compensated if the "true" gyro performance is to be measured.

The testing of inertial quality gyroscopes began at the Charles Stark Draper Laboratory (formerly the MIT Instrumentation Laboratory), in the early fifties. In those days much of the measurement equipment was mechanical and the test data were taken and reduced manually2. Later, with the introduction of more precise instruments, single channel digitized data acquisition systems were used for recording the gyro output signal on punched tape. These data were later entered into a large general purpose computer for off-line reduction of coefficients, stabilities, and residuals. Servo tests were largely replaced by continuous torque-to-balance tumble tests and then by multiple-position torque-to-balance tests. Multiple channel analog recorders were used to gather important information concerning the variations in several environmental disturbance parameters, such as room temperature, gyro tempera­ture, wheel power, excitation levels, etc. Questionable gyro test data could often be correlated with measurable changes in the monitored disturbance parameters. However, the inherent analog representation of these variables required many tedious hours for manual interpretation and did not lend itself to gyro data compensation. Usually, the erroneous portions of the test data were discarded or the entire test had to be repeated. With the increasingly higher costs for precision test equipment and the premium placed on testing manhours, the accumulation of unreliable or unnecessary test data becomes intolerable. Consequently, new techniques for multiplexed digital data acquisition, real-time processing and display have been developed in order to keep the test engineer constantly appraised as to the progress of the test and the usefulness of the accumulating data. In addition, on-line reduction of the test data considerably accelerates the obtaining of final test results. The use of recursive estimation procedures also permits the test engineer to evaluate the optimum time for test termination, thus preventing the accumulation of unnecessary test data.

Automatic computer control of the test specimen position and orientation about all axes, and the control of many disturbance parameters offers even newer possibilities for "dynamic test sequencing". The test mechanization itself can be dynamically altered in real time by the computer program based on accumulated test results. In other words, the test positions and excitation levels would be computed in real time based on an optimization scheme designed to minimize the total test time required to achieve a desired confidence level.

1.2 Purpose

The purpose of this paper is to describe some of the recent advances which have been made in the test and evaluation of precision inertial gyroscopes. The selected material outlines one particular approach to gyro testing and is intended primarily for the exchange and stimulation of ideas. Some of the concepts expressed herein are specifically related to the testing of a single-degree-of-freedom, floated, rate-integrating, magnetically suspended gyro of the type developed by the Charles Stark Draper Laboratory. However, all of the ideas can be applied to any gyro or other incrtial sensing device that requires the same degree of detailed, precise investigation.

In keeping with the purpose of this paper, I have chosen to omit the basic description of a gyroscope. I also assume that the reader has a fundamental knowledge of current test equipment, test techniques and analytical methods. A comprehensive treatment of these subjects has been documented in AGARDograph 128, Inertial Component Testing: Philosophy and Methods, published by Technivision Services, Slough, England in 1970. The information which has been included in this paper is an attempt to extend the referenced materia! in a few selected areas.

1.3 Scope

Gyro testing may be grouped into several categories which are identified by the objectives involved. For discussion purposes these categories may be considered as: (a) acceptance tests, (b) calibration tests, (c) engineering tests, and (d) research and development tests.

Acceptance tests are used in the screening of production line instruments to insure that they meet the minimum requirements for satisfactory field performance. Calibration tests are standard procedures used to establish numerical values for performance equations which describe the essential behavior of a particular instrument. Engineering tests are used to evaluate the gyro's compatibility with system objectives and procedures using a single instrument rather than the three or four gyros that would be required to test a complete system. These tests are designed to produce detailed information concerning the electrical, mechanical, thermal, magnetic, vibration, linear acceleration, and other environments to which the gyro will be exposed when actually installed and operated in a system. Research and development tests are concerned with basic information on the fundamental physical phenomena that may be applied in various possible mechanizations to the class of inertial sensors under investigation.

This paper is addressed to the area of research and development testing of precision gyroscopes. This type of testing in many instances forms the basis for the definition of tests which are conducted in the other three categories. Thus, it is not surprising to find that many of the advanced test techniques and equipment items are initially incorporated into this testing category. Hopefully, some of the concepts which are an outgrowth of research and development testing will set the trend for future testing methods.

Chapter 2 has been organized around the four fundamental phases which are always involved in the testing of any sensor. Some techniques and test equipment requirements are given in the consideration of (i) establishing a suitable and known environment, (ii) applying actuating inputs which are carefully controlled and accurately known in orientation and magnitude, (iii) recording the output responses of the device under test, and (iv) evaluating the results in terms of these experimentally determined output-to-input relationships.

Chapter 3 briefly reviews some of the presently accepted gyro test methods and goes on to describe several novel test techniques which the author has either proposed or tried in recent years. In the conclusions are contained some predictions as to the future trends in gyro testing and some specific recommendations for improving present day techniques.

2. BASIC TEST CONSIDERATIONS

2.1 Overview

During the AGARD Lecture Series on Inertial Component Testing: Philosophy and Methods, presented to the NATO Nations in 1968, a brief description of "future test stations" was given in which digital computers would be used to control and monitor the ongoing tests. Such a test station has been implemented and is currently being refined by the Frank J.Seller Research Laboratory at the USAF Academy, Colorado, USA. The purpose of this new test facility is to conduct in-depth research into the fundamental nature of inertial instrument error phenomena and to develop advanced test techniques, data filtering methods, analysis procedures, and test monitor and display methods for future sensors. In this section the hardware and software aspects of this advanced gyro test station will be described.

The heart of the test station lies in a SEL 81 OB mini-computer as shown in the block diagram of Figure 2.1. This machine is representative of the many 16-bit computers that are currently available at reasonable cost. The CPU memory contains 32,000 words of core storage and has a memory cycle time of 750 nano seconds. A flexible digital input/output system provides the means for interfacing the many specialized test and control units. Under this testing concept the entire test is under full control of the computer. A Hazeltine Alphanumeric Display Terminal (ADT) is the primary operator communication terminal for interfacing with the experiment. A Real Time Executive (RTX) software operating system is used to control the real time tasks in the middle-ground environment and to simultaneously allow data analysis and displays to take place in the background. The operating system itself along with the drivers and handlers for the peripheral devices encompass the foreground environment. A Tektronix Graphics Display Terminal (GDT) is the primary interface device between the experimenter and the data analysis tasks.

For the purpose of discussing the concepts and hardware which have been incorporated into this gyro research testing laboratory, this chapter contains sections covering four functional areas. They are:

(i) The Environment

(ii) Excitation and Control Electronics

(iii) Monitor and Recording

(iv) Data Analysis and Evaluation.

PROXIMITY ELECTRONICS

GYRO

MOTION SENSOR

ISOLATION PLATFORM

CONTROL DEVICES • V

ACTIVE

CONTROL LOOP

BASE PAD

.NVIRONMENT Jt i_ *_

LOW-SPEED DAS

TABLE CONTROL

ELECTRONICS

<—>

B 6700

HARD COPY UNIT

->

GRAPHIC DISPLAY TERMINAL

SEL 810B

C.P.U.

HH

£Z CARD

READER

- J

DISC

UPS

EXCITATION

ELECTRONICS

CONTROL

ELECTRONICS

MONITOR

ELECTRONICS

HIGH-SPEED DAS

COMPUTER

INTERFACE

8-CHANNEL ANALOG RECORDER

ALPHA -NUMERIC DISPLAY

DIGITAL CLOCK

INTERRUPT BOX

MAGNETIC

TAPE ASR 33

LINE

PRINTER

Fig.2.1 Computer-controlled gyro test station block diagram

2.2 Environment

2.2.1 Definitions

The testing of precision inertial gyroscopes requires that the test instrumentation and operating environment be stringently controlled and monitored. Interfering inputs which were once considered negligible have taken on considerable importance in light of the extreme accuracy and repeatability of today's evolving instruments3. For the purpose of the following discussion "environment" is defined as all physical, electrical, mechanical, and propagat­ing parameters external to the gyro case, which may have a direct or indirect effect on its output. Those parameters which are internal to the gyro case will be considered under the section on Excitation.

These external parameters can be divided into three categories: (i) power, (ii) ambient conditions, and (iii) base motion. Power parameters include commercial line voltage, current and frequency, d.c. power supplies, and grounding. Ambient condition parameters include those environmental factors in the immediate vicinity of the test specimen. This includes the room air temperature, temperature gradients, pressure, humidity, magnetic field, dust particles, acoustics, gravity, r.f. interference, and personnel activities. In addition, secondary environmental factors outside the laboratory structure are included for completeness. These are wind speed and direction, solar radiation, precipitation, atmospheric pressure, outside humidity, temperature, water table level, and vehicular traffic. The base motion para­meters include tilt, azimuth orientation, horizontal and vertical translations, and vibration along and about the six degrees-of-freedom of platform motion. These parameters which are used to describe the motion of the test platform can be excited in many ways. Commonly recognized sources include earthquakes, earth tides, microseismic waves, earth tremors and local disturbances, wandering of earth's poles, precession of earth's polar axis, change in speed of earth's rotation, thermal distortion, settling of buildings, direct disturbances by personnel and other cultural noise.

In the next two sections of this Chapter a discussion of the effects of these environmental factors on the testing of precision gyroscopes will be given along with several techniques which should be considered in reducing their influence.

2.2.2 Effects

POWER. The effects of power are often observed during gyro testing, but the exact source is often times elusive and difficult to correct. Sixty cycle noise is frequently present in most low-level signal measurements unless proper grounding and shielding rules are rigidly adhered to. If not reduced to an acceptable level before digital sampling is performed, this high frequency noise component will be aliased into the lower frequency signal region of the data. Section 2.4.3 describes this error effect in more detail. The effects due to power line variations can be observed in the variation of excitation supply outputs, such as the wheel, microsyn and d.c. supplies. Data monitor and recording equipment can also be affected in the same manner.

Perhaps the most detrimental effect of power is caused by an interruption in service which can completely negate all data taken up to that time. Once the excitation to the gyro wheel is interrupted or the gyro's temperature control system is temporarily terminated, irreversible changes take place in the state of the instrument. This nonlinear change will prevent linear analysis, modeling or correlation between time related phenomena. j*Vlso, a test which is under computer control can present some very serious problems to the test operator whenever line power is terminated, interrupted, or experiences severe transients if preprogrammed shutdown and recovery procedures are not implemented.

AMBIENT CONDITIONS. The effects of ambient environmental conditions on gyro testing operations are almost too numerous to mention. Many of these effects are of a secondary nature in that they create test errors by causing a change in some other parameter. For instance, room temperature is one of the worst offenders. It affects almost everything else of importance. All electronic components have some degree of temperature sensitivity and therefore must be very carefully regulated. The most critical items appear to be the wheel supply, signal and torque generator excitation sources, gyro torque loop readout electronics and precision temperature monitor electronics. The gyro temperature distribution and input heater power requirements are both affected to some extent by room temperature variations in spite of careful insulation and a thermally controlled test jacket.

The dimensions of the gyro holding fixture and the test table positioning fixture are also controlled by the room temperature. Slight variations will cause tilts and rotation rates about the gyro's input axis which will be imbedded in its output signal. These error effects are almost impossible to correct since almost all of the base motion sensors are located on the stationary structure of the test fixture or on the supporting platform. Precision base motion sensors which are required to measure in the milliarcsecond region are predominantly influenced by the surrounding air and mounting base temperature. Room temperature is also an extremely difficult parameter to compensate for since it is a specially distributed quantity which affects numerous devices at different locations in many different ways. Consequently, any improvement that can be made in the reliability and control of the test laboratory temperature environment will greatly reduce the requirement for special temperature-controlled enclosures and add immeasurably to the accuracy of any specific test.

The other environmental parameters in this category will have a greater or lesser effect on the operation of a gyro depending on the nature of the particular facility and the test being conducted. For example, variations in room pressure will have a very significant effect on the base motion environment if the test platform is supported by a passive pneumatic isolation system. The humidity level can affect the mutual conductance between signal lines and the buildup of electrostatic charges. This can create some peculiar problems in the operation of a computer-controlled test setup. Time varying magnetic fields generated by large current carrying lines or electromagnetic devices are generally troublesome when they affect low-level signal lines. The gyro and the table mounted electronics can be influenced by a static magnetic field since they are usually rotated with respect to an earth reference frame during test operations. Selective magnetic shielding is almost always required to reduce this problem. The effect of gravity is mostly felt in the deflection of the table and gyro holding fixture when oriented in different positions with respect to the local gravity vector.

Acoustic noise is a form of vibration that may affect both the gyro and personnel working in the area. Recently a new gyro test laboratory was built with laminar air flow across the room. Due to the particular shape of the air ducts which were used, a low-frequency drumming sound which was objectionable to workers in the room was created. The correct combination of baffling, vibration isolators and a reduced air flow was required to correct the situation. Electronic cabinet cooling fans and motors in computer peripheral equipment can also raise the acoustic noise level to a point where it is fatiguing to personnel working in the area. Personnel activity in the immediate vicinity of the gyro under test can create asymmetrical floor loading, disturb air flow patterns and generate moving warm spots in an otherwise settled test environment.

External environmental conditions will most often generate lesser internal disturbances, which in turn cause test errors to occur. For example, the thermal distortion of buildings due to solar radiation varies with the building dimensions, materials of construction, and the character of the foundations. Daily tilts of a test lab floor have been correlated with the amount of solar radiation to which the building is exposed on any given day2. Wind loading on the side of a building can also generate a marked increase in test floor vibration level. The tilt level of the lab test floor has also been related to outside temperature3, atmospheric pressure4, water table level5, and vehicular traffic*.

BASE MOTION. The calibration and testing of precision inertial gyroscopes is currently limited by translational and angular vibrations and long-term tilts of the test platform on which the instrument is tested. Small motions of the test platforms will change the orientation of the gyro with respect to the primary input quantities, earth rate and gravity. However, the largest errors in practice in gyroscope testing are caused by the rates associated with angular vibration of the test platform. The magnitude of this effect will depend on the frequency of the input and the amount of time over which the gyro data is averaged.

The errors due to an uncertainty in the alignment of the gyro depend on its orientation with respect to the earth rate vector. The error is a maximum when the gyro is oriented with its sensitive axis perpendicular to the earth axis, a fairly common test orientation. The gyro drift rate can be calculated from Equation (2.1),

E, = W,Esin0 , (2.1)

where W-g is the angular velocity of the earth relative to inertial space; 0 is the misalignment angle.

The sensitivity of E, to 0 is then given in Equation (2.2)

3E. _ —- = 0.005 meru/sec . (2.2) 30

Thus, a test platform misalignment of only 0.2 sec is permissible if the test error is not to exceed 0.001 meru.

The drift rate error caused by a misalignment with respect to the gravity vector is negligibly small compared to the earth rate vector error calculated above. A misalignment angle of 0.2 sec causes a gravity input uncertainty of 10"6 g. When this is multiplied by a gyro drift coefficient as high as 10 meru/g, an output uncertainty of only 1CT5

meru results. The sensitivity is expressed as follows:

8E —1 = (5 xl(T6)D meru/sec (2.3) 30

where D is expressed in meru/g.

Since the gyroscope is a rate-sensing device, it will be most sensitive to angular oscillations of the test platform about its input axis. For an angular vibration displacement given by

0 = A sin wt (2.4)

there is an angular velocity of

0 = Aw cos wt . (2.5)

The gyro will sense a maximum rate for an angular vibration of amplitude A and frequency f of:

0max = A2vTf. (2.6)

Thus, for a 1 arc second angular oscillation at 0.1 Hz, the instrument will sense an error rate of 0.628 sec/sec or about 42 meru. This sensitivity is expressed as

3E = 420 f meru/sec (2.7) 9 t 9vib

where f is expressed in Hz.

In testing a gyro it is common practice to pass the data through a filter or to average the data for a period of time which is presumed to be long compared to the disturbance inputs to the instrument7. If the filter has the characteristics of a first-order system:

A ou l = • AIn 1 + TS

(2.8)

where T is the filter time constant (seconds) and s is the Laplace transform variable. The indicated gyro rate is

27Tf0 0 =

| l + (27rfr)2]1/2 (2.9)

The indicated error in drift rate due to one milliarcsecond (0.001 sec) of angular vibration for several averaging times as a function of vibration frequency is plotted in Figure 2.2(a). At frequencies below 1 cycle/day, the error is primarily due to the misalignment relative to earth rate. For a 10-second time constant filter (20-second averaging time), the error between 1 cycle/day and 1 cycle/minute is due to the ability of the gyro to measure the instantaneous rate associated with the undesired oscillation. At frequencies above 1 cycle/minute, the error is the uncertainty in position error divided by the averaging time. If the platform oscillations can be held within 0.001 sec at all frequencies, then a first order filter with a cutoff frequency of 0.0023 Hz should reduce the measured drift rate errors to 0.001 meru. A 0.0023 Hz filter requires that the gyro test data be averaged for about 3.5 minutes before it reaches 95% of its final value.

o

<JJ

£

•r— 1 -

10

1 cycle/day 1 cycle/hr

I t • t 1 cycle/min

10 -5 10 -4

—I H 10"3 10"

Frequency (Hz)

1 1 cycle/sec

10

Fig.2.2(a) Maximum error in gyroscope testing due to 0.001 arc second of angular vibration

It would appear from the above discussion that a long averaging time is desirable in gyro testing. However, the time required for an instrument calibration is directly proportional to the averaging time. In addition, high-frequency noise signals may produce saturation or other undesirable effects. Not only does the test become very long and expensive, but it is not possible to obtain any information on the short term drift characteristics of a gyroscope when heavy data filtering is used to reduce the effect of unwanted input vibrations. For example, in tumbling tests a common filter cutoff frequency is 0.01 Hz.

Another way of looking at these same requirements is to first determine what the upper frequency is for the useful data. This will determine the cutoff frequency of the filter that can be used. Then using the error sensitivity equations (2.2) and (2.7), the requirement for the test platform stability can be determined.

Figure 2.2(b) shows the platform stability requirements for taking up to 1 Hz of useful gyro data if the test error is to be no longer than 0.001 meru. This plot shows that platform vibration stabilities approaching 10~6 sec are required if accurate data is to be taken up to 1.0 Hz. However, a long-term platform stability (longer than one day) of only 0.2 sec is needed to meet this same requirement.

Earth Rate Coupling Error

10

1 cycle/day 1 cycle/hr 1 cycle/min

J 1 cycle/sec

_J_ 10 10 10"3 10'2 10"

Frequency (Hz)

Fig.2.2(b) Requirements for test platform stability for limiting drift rate error to 0.001 meru

Gyroscope errors due to translational vibrations are generally negligible compared to the rate errors previously discussed. Assuming that the test instrument has a drift coefficient of 10 meru/g, then an input vibration level of 10"4 g would be required to create an error of 0.001 meru. A gyro also has a sensitivity to the square of the applied acceleration, often called compliance or anisoelasticity torque. Assuming a drift rate sensitivity of 10 meru/g2, a vibration of 0.01 g would be required to produce a drift rate error of 0.001 meru. These acceleration levels are considerably larger than those that can be achieved in a well controlled laboratory seismic environment.

2.2.3 Techniques

POWER. Little can be done to control commercial power variations and frequent interruptions. Changing loads within the laboratory building and in the surrounding community change the input voltage levels. Electrical storms cause intolerable interruptions which can only be corrected by maintaining a backup generator or batteries on the line at all times. The most effective means which has been found for correcting these problems is to run all of the critical electronics and the computer mainframe from a set of Uninterruptable Power Supplies (UPS). These battery supplies contain a 60 cycle inverter and a synchronized 60 cycle clock in order to supply the required continuous, highly-regulated line power. The batteries are constantly recharged by the commercial line power. In the event of a power failure, the UPS will deliver full rated power for a period of 15 minutes and then will gradually diminish its output. This will allow the test operator sufficient time to turn off the less important electronic units in an organized manner and will allow sufficient time for a gasoline-powered emergency power system to be activated. Power monitor equipment should be installed to detect and record both the input and UPS power for fluctuations, spikes and dropouts. This information can be correlated with the other recorded variables to assess the impact that it has on gyro testing operations.

A separate earth ground system installed in the laboratory has been found to be useful in the elimination of unwanted noise in low-level signal cables. Great care must be taken however to insure that no large current carrying return lines are connected to this ground, or else the desired isolation advantage will be lost. Special attention must be paid to following acceptable grounding and shielding rules if adequate noise immunity is to be achieved on monitor signal lines8.

AMBIENT CONDITIONS. Three techniques arc available for reducing the effect of environmental disturbance parameters. First, the level of the input disturbance can be reduced. Second, the instrument can be shielded or isolated to provide some degree of immunity from its environment. Third, the variations in the disturbance parameters can be monitored and this data used to compensate for the error components. All three of these techniques must be used if meaningful tests are to be conducted on precision gyroscopes.

Obviously very little can be done to reduce the variation or level of external environmental factors, such as temperature, humidity, pressure, vibration, wind, solar radiation, etc., except for selecting the best possible test site. A well-constructed building will provide some degree of isolation from these elements, but closed-loop control of the laboratory environment is still essential. A good air conditioning system should be able to reliably provide ±1.0°F temperature control of the entire room, 30-50% humidity control, and class 10,000 cleanliness. Sound absorbing walls and partitions between laboratory personnel and the gyro test platform will substantially reduce acoustic vibration and thermal gradient variations. Depending on the degree of sensitivity of the gyro to magnetic fields, some degree of magnetic shielding is always required. Usually the gyro will include some magnetic shielding as a part of its outer housing or thermal jacket. Additional shielding may have to be provided on the holding fixture, especially if the gyro proximity electronics generate any large magnetic fields.

After the background environment has been controlled to a level which is reasonably consistent with design practices, capabilities and cost, all of the potential error generating disturbance parameters should be monitored and synchronously recorded along with the gyro test data. Once the sensitivity of the gyro to environmental and subordinate inputs has been determined, data correction or compensation can be accomplished. This technique is described more thoroughly in Section 3.2 on Gyro Error Modeling. Also, by identifying which parameters generate the largest error effects, a concerted effort can be made to reduce the level of that disturbing input. Thus, more knowledgeable tradeoffs between which parameters should be controlled better and between the degree of control and monitor to be employed become possible.

BASE MOTION. Most of the techniques which have been used to isolate inertial sensor test platforms from the surrounding noise environment can be divided into three categories. These are (i) bedrock attachments, (ii) passive isolation, and (iii) active isolation.

Bedrock: In seismically quiet regions of the earth, one fairly successful method of achieving low platform tilt and vibration levels is to attach the test platform directly to bedrock. Of course, this is only possible in areas where significant outcropping of bedrock is available. At the Northrop facility in Norwood, Mass., several test stands have been successfully hard-coupled to the basement rock9. These stands behave as rigid blocks and do not isolate from regional sources. In this particular area the regional sources were measured to be less than 1 ug r.m.s. over the 0.1 to 10 Hz band. Rotations were found to be much less than 0.1 arc second in this same band.

The main advantage of the bedrock structure is that it can provide considerable isolation of the test stand from floor motion which is generated by test personnel and vibrating machinery. For example, the ratio of the motion of the stand to that of the floor at the Northrop facility is 220 near 10 Hz. The ratio is even larger throughout the remaining frequency band. Although this isolation technique is suitable for some regional areas with a competent rock structure, it is not applicable to all sites. Similar stands bonded to a low velocity, compliant material would not move in unison with the basement. The result would be a substantial enhancement of stand motion over the ground and a partial conversion of linear motion into a rotation10. Therefore other techniques have been developed to isolate the test stand against base motion disturbances.

Passive Isolation: Most of the softly sprung, passive isolation systems in existence can be placed into two categories: those using mechanical springs and those supported on air columns. The Aerospace Guidance and Metrology Center, located at Heath, Ohio, has mechanized both approaches. The first consists of a test floor structure suspended by a pendulum rod from an isolating suspension system at the top of a 25-foot A-frame. The isolation system consists of a beryllium-copper beam and toggle assembly which supports a 20,000 lb test platform and provides a nonlinear soft spring for the vertical axis. Horizontal isolation is provided by the 24-foot-long pendulum rod".

Many of the pneumatic isolation systems in use for inertial sensor testing employ mechanically-actuated servo valves. Three of these valves can maintain a reference surface on the platform fixed with respect to the supporting base pad. This is accomplished by allowing pressurized air to either enter or exhaust from the supporting cylinders. Unlike a mechanical spring this allows for a zero static deflection under static load conditions.

Active Isolation: Achievement of platform isolation below the suspension natural frequency must be accom­plished by active means. Active control of a mechanical spring support has been accomplished at Stanford Univer­sity12. This system is actuated by two solenoid-type electromagnets which apply up to 20% of the bias force to bellville type table support springs. An all-pneumatic system was developed for the Heath facility13. This system uses a large aluminum pendulum which controls the flow of air through two nozzles impinging on the pendulum. This differential pressure, which is proportional to platform tilt, is used to control the pressure in the supporting cylinders. Another approach, to be described in this section, uses electromechanical level sensors. Their outputs are used in a servo feedback loop to drive electrical-to-pressure transducers which, in turn, control the cylinder pressures.

An alternate scheme for a tilt isolation platform was built at the MIT Draper Laboratory14. In this system the higher frequency passive isolation supports were not used, since it is the low-frequency angular motion (ground tilt)] which has the most serious error effect in gyro testing. Instead, the platform was mounted to the supporting structure through highly refined screwdriven hydraulic jacks called micromotion drives. Two micromotion drives are actuated by electric servo motors. When a ground tilt occurs, electronic tilt sensors are displaced from null. This displacement causes the servo system to act in such a way as to maintain the platform in a level position. Later improvements to this approach15 incorporate a gyroscope to control a micromotion drive with a much greater frequency response (25 Hz). The tiltmeter signal is used to drive the gyro torquer in order to eliminate long-term drift from affecting the platform level.

Isolation Platform: The isolation platform at the USAF Academy is constructed of steel reinforced concrete. It appears as a 25 foot square from the top with nine circular piers rising 2.5 feet up from this surface and protruding through a false floor. The bottom has a cruciform shape that is 4.5 feet high. The total platform weight of 450,000 lbs is supported by twenty undamped pneumatic isolators and floated approximately 1*4 inch above the base slab. Reference 16 contains a more complete description of this facility.

The choice of a soft-sprung supporting structure or a rigid ground coupling depends to a great extent on the application. For many experiments, where high frequency accelerations can excite natural frequencies in the test devices, the low-natural-frequency, passive system is to be preferred. In this case the low frequency motions of the platform are of little importance since they do not impart any appreciable accelerations to the item under evaluation. In the case of inertial component testing, however, these small-amplitude, low-frequency motions are intolerable. In gyro testing, the tilt and orientation angles and angular rates are of greatest importance, while the tilt angles and linear accelerations are of greatest importance in accelerometer testing.

A soft-sprung passive isolation system does an excellent job of eliminating ground motion of the base slab above the suspension natural frequency. The amplitude attenuation behaves essentially as a second order low-pass filter as shown in Figure 2.3. Transmissibility at low frequency is 1. The price that must be paid for achieving this isolation from base motion, however, is a high susceptibility to disturbance forces on the platform itself. This is shown in Figure 2.4. The transmissibility of force disturbances at low frequency is equal to 1/K , where K is the stiffness of the supporting springs.

Since o>n = [K/M]1/2 , lowering the platform natural frequency increases the platform sensitivity to force disturbances by 1/w2, . The desired natural frequency of the passive isolation system then amounts to a trade-off between the amount of isolation against base motion versus force disturbances that is desired. A useful criterion would be to limit the platform motion which results from typical force disturbances to a value equal to the expected base motion at low frequencies. This will essentially keep the platform motion which results from both inputs equal at zero frequency.

The USAF Academy platform has a K = 580,000 lbs/ft and a natural frequency of 1 Hz. For this system, a motion sensitivity of approximately 20uinches/lb results.

During testing operations, no personnel activity occurs on or around the isolation platform on which the test table is mounted. Consequently, the largest disturbance forces to the platform will be caused by atmospheric pressure changes within the room. This results in a change in buoyancy force exerted on the platform. For a platform having a volume of 3100 cubic feet and a 10% variation in the density of air that is nominally 0.075 lb/ft3,

10

o.i 1.0

Frequency Ratio u/u

Fig.2.3 Isolation to base motion inputs

a force variation of about 23 lbs is imposed on the platform. This results in a deflection of 460uinches. This effect is equivalent to a I arcsecond tilt of the base slab which might occur during testing.

Motion Sensors: In order to sense the motion of the platform and separate all six degrees-of-freedom in the frequency band of interest (0-10 Hz) a combination of sensing devices is required. For the very low frequency rotations (0-0.1 Hz) a high quality, two-axis tiltmeter is used. The azimuth orientation angle is measured with a four-position gyro-compassing system with considerably less accuracy than the tiltmeters. However, it does provide a continuous automatic azimuth determination updated every four minutes. Digital estimation techniques are used to improve this accuracy over longer time periods. No provision is made to measure the low frequency (d.c.-0.1 Hz) translational motions, except for the vertical motion with respect to the base slab. This is measured with a resolution of about lOuinches by means of four proximity capacitive sensors. Since neither gyros nor accelerometers are sensitive to this low frequency, low amplitude linear motion, it is of no consequence.

The higher frequency motions (0.1-10 Hz) of the platform are sensed by an array of short-period, matched seismometers. These are positioned on the platform as shown in Figure 2.5. The output of each seismometer can be represented to first order as follows:

V, = z + wxd

V, =

V, =

2 - wxd

z — wyd

• « ,

= z + wyd

= X + W?d + Wyh + Oyg

(2.10) cont'd

11

H,

H,

•y + wzd + w xh — 0xg

x + wzd — w„h — 0yg (2.10) cont'd

H4 = y + wzd - wxh + 0xg-

From the above equations, all six degrees-of-freedom of platform motion can be determined from Equation (2.11).

z = J(V, + V2 + V3 + V4)

Wy = V 4 - V 3

2d

wz = 4(H, + H2 + H3 + H4)

*-H-^-(V-^)-^/K-v ' )dt2

(2.11)

1000

QJ

u

CVI Q

s-o

u ro

ro

Frequency

Fig.2.4 Isolation to disturbance forces

i—m-r

Fig.2.5 Orientation of seismometers

Servo Control Devices: In order to remove the effects of base motion and force disturbances on the platform, a means of applying a controlled force to the platform is required. This has been accomplished with two separate systems. First, the pressure in the supporting air cylinders themselves is varied in order to remove the low frequency transients (below 1 Hz). Figure 2.6 depicts the main components used in this system. A push-pull arrangement is used in order to keep the applied rotations about the center of gravity of the platform. In addition, the use of two electrical-to-pressure transducers, regulators, and boosters increases the amount of air that can be exchanged, and thus increases the speed of its response.

In order to obtain a stable closed loop servo operation, the resonant response of the platform must be suitably damped. Although a means is available to provide some damping through the use of a small orifice between the upper and lower cylinder, this can only provide a damping ratio of 0.6 when set at the optimum position17. An even more serious limitation results from the fact that there is no way that the damping values can be set equally to provide the same damping to all cylinders. Consequently, each of the twenty cylinders ends up with a different damping ratio and damped natural frequency. All of these different spring-mass systems then generate considerable cross-coupling of the platform motion between the two tilt axes. The use of dashpots in parallel with the spring suspension offered another possibility. However, at high frequencies the mass would be coupled to the earth more closely through the dashpot than through the spring, thereby reducing the effective isolation. Damping adjustments would still be difficult and selective damping across the resonant frequency band would not be possible.

To easily adjust the amount of damping and to control the frequency region over which it acts, an active damping mechanism was added to the system. Figure 2.7 depicts the essential components of this damping loop. Four vertical seismometers and four small electrodynamic shakers are used to actively damp platform rotations about the two horizontal axes and translations along the vertical axis. Four additional shakers act horizontally at the platform center of gravity. These shakers are controlled by the four horizontal seismometers and provide damping for rotations about the vertical axis and for translations along the two horizontal axes. This arrangement is shown in Figure 2.8.

The eight seismometers are compensated so that their output voltage is proportional to case accelerations in the frequency band from 0.1 to 10Hz (Ref. 18). Due to the extremely large mass (platform) that is attached to the moving element of the shaker, its natural frequency is quite low (<0.01 Hz). Therefore, its output force falls as 1/f in the frequency band in which it is used (0.1 to 10 Hz). This resulting integration generates a force that is proportional to platform velocity when acceleration is used as the control signal. Adjustment of the shaker amplifier gain controls the amount of damping force that is applied to the platform.

13

ELECTRICAL TO PRESSURE TRANSDUCER

I PARTIALLY SUPPORTED REGULATOR

I FLOW

BOOSTER

< = 3 25psi

<£=3 75psi

25psi

75psicz^

120psi n 120ps !<-=•>

TILT SENSORS

SEISMIC PLATFORM

ELECTRICAL TO PRESSURE TRANSDUCER

I PARTIALLY SUPPORTED REGULATOR

I FLOW

BOOSTER

£* AIR CYLINDER

77777 W?7^////)777777/ N

--OrJ-J I I I mJ-mm.

AIR CYLINDER

/ / / ' /

Fig. 2.6 Tilt sensor control loop

2.3 Excitation

2.3.1 Test Table

A two-axis test table is mounted on one of the test pads of the seismic isolation block. This system provides a flexible, highly versatile and accurate test fixture which is used for testing, alignment and calibration of precision gyroscopes or accelerometers. The table contains d.c. torquers for the primary and secondary axis drives, high-accuracy angle readout systems, sliprings, precision bearings and a two-way adjustable gyro alignment plate. A three-bay console of table control electronics contains the servo controllers, auxiliary rate and power control amplifiers, angle encoder displays, a computer interface and display terminal and a phase-locked loop position servo unit.

This table assembly can accept a test package weighing up to 25 lbs with a size of 7-inch diameter by 16-inch length. The primary and secondary axes are driven by direct-mounted torque motors in both clockwise and counter clockwise directions. The secondary axis is built as a self-supporting subassembly, permanently mounted to the 24-inch diameter table top. The secondary axis assembly is supported by precision ball bearings. On this axis is mounted a precision two axis gyro alignment plate. This is used to accurately position the gyro's input axis to within one arcsecond of the secondary axis of rotation. It can also be used to manually reorient the gyro with respect to the fixture axes. The primary axis is mounted on an aerostatic air bearing which virtually eliminates bearing friction. This axis of rotation is gimballed around the table tilt axis which permits accurate manual positioning of the table axis about a horizontal trunnion axis. This trunnion axis is supported on precision ball bearings in a yoke. The yoke in turn is mounted on the base which incorporates an alignment capability to permit

14

DRIVE

AMPLIFIER

DRIVE

AMPLIFIER

COMPENSATION

NETWORKS

J " ^

a ] C ] _ £

SEISMIC PLATFORM

SHAKER

\\i\\mm AIR

CYLINDER *; s

//77mm9y////777777)r/

AIR

CYLINDER

R r SHAKER

TV Fig.2.7 Seismometer controlled damping loop

proper orientation of the trunnion axis in azimuth and relative to gravity. A set of attachable, stationary-mounted counter weights on the tilt axis gimbal are used to balance the primary and secondary axis fixture assemblies about the tilt axis. Additional small counter weights can be bolted to the outer ribs on either or both sides of the secondary axis yoke structure. These are used to balance the table about the primary axis. Moveable weights mounted on the secondary gimbal balance the secondary axis. Smaller counter-weights are contained in the gyro table mounted electronics to balance the gyro alignment plate and test package about two axes.

The table is bolted to a steel template which is cast in the concrete seismic block. The base alignment device permits a ±5 degree fine adjustment in azimuth at any location of the table azimuth. Leveling of the base with a resolution of ±1 arc second is achieved by means of three screws. Thus, only three points are used to transmit forces to the base.

The orthogonality between the tilt and the primary axis is less than 2 arc seconds. The orthogonality between the secondary and the primary axes is within 0.5 arc second. A 720-pole encoder is mounted on each axis and has an absolute accuracy of ±1 arc second with a 360° resolution of 0.36 arc second. Both axes are capable of being driven from 0.25 G/sec to 1200°/sec ± 0.25% in the auxiliary rate mode. The primary and secondary axes can be commanded to move to any position at any desired rate up to 1000 times earth rate under computer control. Once positioned, the phase-locked loop servo electronics hold the table axes to within 0.1 arc second. Great flexibility in the operation of the table is achieved by its continuous monitor and computer control. For instance, any position, rate, acceleration/deceleration profile, and dwell time can be commanded independently and simultaneously to both of the controlled table axes. The table rates and positions as a function of time are continuously monitored by the computer and recorded for later analysis and interpretation. Small amplitude, low-frequency oscillations can also be programmed into both axes to investigate gyro frequency response or angular rectification effects.

2.3.2 Gyro Electronics

Figure 2.9 is a drawing which illustrates the complement of electronics equipment which is used to excite, control and monitor the gyroscope under test. Although this test station has been customized for a particular gyro, the excitation functions are general in nature and can be adopted to other instruments. The main improvements

IS

Y

A

DRIVE AMP

•

y

I COMPENSATION

NETWORK

n / I • -1

SHAKER

COMPENSATION NETWORK

I DRIVE AMP

/ < / • / / / / . /

'<L

SHAKER DRIVE AMP

I COMPENSATION

NETWORK

H SHAKER •

COMPENSATION NETWORK

I /

DRIVE AMP

Fig.2.8 Horizontal damping loops

in this station over previously used electronics is in the use of computer controlled excitation sources and the addition of many monitor and signal conditioning units.

The computer operated gyro excitation sources include a two-phase wheel supply, signal generator microsyn supply, torque generator microsyn supply, a magnetic suspension microsyn supply and two d.c. power supplies. These units are controlled by the computer in terms of their operating voltage, frequency and mode (off, standby, slow rise, run etc.). Thus, not only does the computer control the on and off sequence of the test specimen, but it can also generate precise variations in the excitation levels in order that the instrument's sensitivity to these input excitation parameters can be investigated.

The control functions are handled by those electronics which are essential to the closed-loop operation of the many controlled test parameters. These include the gyro zone-temperature controllers, rate-feedback servo amplifier, and float position control circuits.

Six separate controllers are used to control the temperature of the gyro at its six cardinal points: along its positive and negative spin axis, along its positive and negative input axis and at each end of its output axis. The average gyro temperature and the temperature gradients along the principal gyro axes are regulated by the individual controllers, but the set points can be changed by the computer. This is accomplished by switching the value of the temperature bridge resistors used in the controller. At least five additional temperature controllers are used to regulate the temperature of the gyro thermal jacket. The remaining five temperature controllers are used to control small enclosures in which several tilt and seismic vibration sensors are mounted.

The individual electronic cabinets are temperature controlled by means of a sensor, controller and heating element. The incoming air to each rack is separately heated by controlling the input voltage to a calrod unit which is inserted into the air stream. A long term stability of 0.25°F is possible using this scheme.

The position of the gyro float relative to its case is controlled by a passive magnetic suspension network. The null points for this passive circuitry are determined by the value of the suspension tuning capacitors. By switching the values of these capacitors under computer operations, the relative position of the gyro float can be controlled. This technique is essential in the investigation of gyro error torques which result from electromagnetic and fluid forces which are set up by small float motions.

1(1

(5) TEMP

CONTROLLERS

(8) TEMP

MONITORS

FLOAT

POSITION

MONITOR

SIGNAL

SELECTOR

HIGH SPEED DATA

ACQUISITION SYSTEM

(16) PRGM,

LOW-PASS

AMPLIFIERS

(16) PRGM.

LOW-PASS

AMPLIFIERS

(5) TEMP

CONTROLLERS

(8) TEMP

MONITORS

GYRO

SERVO

AMPLIFIER

THREE-POLE

LOW-PASS FILTER

S.G,

SUPPLY

(16) PRGM.

LOW-PASS

AMPLIFIERS

(16) PRGM.

LOW-PASS

AMPLIFIERS

LINE REGULATOR

MULTI­FUNCTION METER

PHASE

ANGLE VOLTMETER

SCOPE

COMPUTER

INPUT/

OUTPUT MULTIPLEXER

T.G.

SUPPLY

WATTMETER

WHEEL

SUPPLY

LINE REGULATOR

MASTER

FREQUENCY

SOURCE

DISTRUBTION

GYRO

TEMPERATURE

MONITOR

DC POWER

SUPPLIES

SUSP.

SUPPLY

PRGM. DC

UPS

SYNCHRONIZEF

BATTERY

PACK

UPS

4 A

UPS

<r B

UPS

4> C

BATTERY

PACK

BATTERY

PACK

Fig. 2.9 Gyro excitation electronics console

A gyro servo amplifier is used to maintain the gyro output axis rotation very close to a zero position by supplying a rate-feedback current which is proportional to the summation of all torques on the gyro float. The signal generator secondary signal is amplified, demodulated, compensated, modulated, and used to drive one winding of the gyro torque generator. A d.c. output voltage, proportional to the command torque applied to the gyro float, is also provided by this unit for monitor purposes.

The master frequency source provides stable, hardened and synchronized frequencies to all a.c. excited units. A pair of Fluke d.c. computer-controlled power supplies are included to conduct parameter sensitivity tests on d.c. powered units. Two solid-state line regulators are used to provide an additional level of line power regulation to all of the critical electronic units, such as the excitation sources and precision monitor equipment. The entire test console is run off the unintcrruptable power supplies for regulation and reliability. The computer input/output multiplexer receives commands from the control computer on a single input/output channel and routes the commands to one of sixteen controllable units. This interface unit also provides the capability to monitor and to input manual commands to the excitation sources. This is a very important requirement during software development and also provides a backup capability in the event of a computer breakdown.

The remaining equipment items in the gyro electronics console are used for data monitor, described in the Monitor Section of this Chapter.

They are therefore

2.3.3 Seismic Simulation

The base motion isolation system which has been described in Section 2.2.3 of this chapter has a wider application than merely removing undesirable input motions. It can also be used as a six degree-of-freedom seismic motion simulation facility. Any seismic level motion spectra that has been recorded or that can be generated by the computer can be applied to the gyro under evaluation. This capability makes possible the investigation of gyro error terms which are the result of small input tilts and/or vibrations. Being able to apply controlled input levels independently about a single axis is extremely helpful if one wishes to separate the individual vibration error terms.

17

This seismic simulation facility also has the unique ability for optimizing a gyro calibration scheme in terms of positions and data sampling techniques for a particular operational or test environment. Once the desired base motion has been monitored using an array of seismic sensors and recorded on magnetic tape it can be played into the control devices which regulate the motion of this isolation platform. This motion can be repeated as many times as necessary until the best gyro calibration procedure has been determined for a particular gyro in that base motion environment. Thus, this unique capability provides another form of input excitation to the gyro test specimen.

2.4 Monitor

2.4.1 Background

In the evaluation of precision inertial guidance components and systems, great pains must be taken to insure that the test equipment, instrumentation, and test techniques are sufficiently accurate as to not affect the validity of the test data. In the past, this task has been mainly accomplished by periodically checking the accuracy and calibration of excitation and monitor electronics, mechanical alignments and stability, and by scanning analog recorder traces of critical controlled variables. These procedures can often lead to the identification of disturbances which could affect the measured performance of the specimen under test. These disturbances can then be accounted for by selective data editing or by repeating the entire test sequence under more favorable test conditions.

The increased use of analog-to-digital converters, multiplexing equipment, and tape storage devices within the test laboratory has permitted a wider spectrum of signals to be monitored and recorded for later analysis. This post-analysis is generally accomplished on a large-scale, general purpose digital processor. The results of several of these studies have quantitatively confirmed inter-relationships between variables that were suspected for some time to exist and have even revealed new relationships not previously known to exist. Correlation studies, although widely used in other fields, have recently gained popularity in the study and examination of inertial guidance test data.

The amount of test data of this nature that has been analyzed using these techniques has been limited to a few instruments and a few selected variables. The reason for this limited investigation is due in part to the extensive problems of data formatting, coding, intermediate storage, decoding, editing, and processing on remote computer installations.

Long tum-around times have two serious disadvantages in research and development testing. These are (i) that the test electronics and instrumentation must remain idle during data processing, analysis and interpretation to determine the validity of the data and the appropriateness of the test method used and (ii) long test sequences are carried to completion only to determine during later analysis that the test must be reaccomplished due to faulty equipment or test technique. These two factors alone have accounted for considerable inefficiency in the use of extremely expensive test equipment and manpower. Unreasonable delays in completing the basic test objectives for a single test specimen necessitate the use of multiple test stations in order that simultaneous test programs can be conducted, since much idle time is spent on each station. Reliability and maintenance of equipment can then become a burdensome problem.

The purpose of this section is to outline the methods that can be used to streamline this acquisition, processing, and display of test data and to minimize the amount of wasted test and analysis time. The requirements for a small, flexible, multi-purpose digital computing system within the laboratory to monitor, control, and analyze are developed. An outline of the basic software programs required for a research oriented test station is given in the next section in order that the programming task and hardware capabilities can be assessed.

2.4.2 Data Acquisition

There are two different approaches that can be taken to the data acquisition task. The first of these is to use a relatively low-speed analog-to-digital converter and scanner. This system uses a multi-function meter that has considerable flexibility in selecting the signal range, function, input impedance, and delay time. It acts as a universal signal conditioner in that a.c. voltages, d.c. voltages, resistances, and frequencies can be monitored having an extremely wide range of input levels. For this increased flexibility and compactness one must sacrifice speed. In many cases, such as in the collection of long-term tilt, temperature, pressure, or power fluctuations, data sampling speed is not a problem and the low-speed system is advantageous. On the other hand, however, low sampling rates for long time periods necessitate the use of analog filter circuits with very long time constants as will be explained later.

Such circuits may be impractical to construct, and neglect of this factor may cause substantial measurement errors leading to erroneous conclusions concerning the data. In addition, if each data record cannot realistically be considered to be an instantaneous time slice, then skewing of the data will occur and later correlation analyses may misrepresent cause and effect relationships in the data. In those cases where infrequent samples are to be taken under essentially steady state conditions, such as a fixed position tumble test where sufficient settling time is allowed, a low-speed system can provide all of the required capability with the least complexity and cost. Also, better signal resolution can ultimately be achieved in such a system.

18

In those testing situations where an analysis of the random characteristics of the monitored data is desired or where the measured signals are changing value at a frequency greater than about 0.1 Hz, a high-speed data acquisition is more appropriate. A high-speed system can accommodate a wide range of input signal levels (e.g. lOmv to 10 volts), but is restricted to measuring d.c. signals only. Throughput rates of 20,000 samples per second are easily achieved. This will handle 1000 records per second for 20 channels or 100 records per second for 200 channels which should be adequate for most laboratory applications. If considerably slower sampling rates are used, the data records still encompass the short sampling interval and data skewing is of no significance. This will allow considerably more time for the computer to make intermediate calculations between data records. Digital filters can then be imple­mented to cut the effective sampling rate down considerably without the requirement for the long time constant analog filters. Use of a high-speed system will allow the test engineer to measure and investigate the noise character­istics of various signals, the higher frequency content and transient effects in gyro torque currents, and other short-term phenomena. During fixed position testing, the signal noise statistics in each position can be calculated and used to weight the data during later analyses. Also, the signal variations during fixture repositioning can be recorded in order that transient and settling effects can be investigated. Removing the constraints of heavy analog filters and settled signal measurements, gives the test engineer considerably more freedom in designing parameter variation tests or reducing the total test time required to gather statistically significant measurements. The limita­tions inherent in a high-speed data collection system can be overcome through the use of input preamplifiers, signal conditioners, and filters for each data channel. This means that a separate a.c. to d.c. converter, resistance measure­ment bridge or frequency converter must be provided for anything other than d.c. signals.

Except for the greater complexity and cost of the high-speed data acquisition system, it can be made to perform all of the functions of the low-speed system while retaining all of its flexibility and accuracy. Increased resolution can be obtained by software manipulation. Since the added speed of measurement can be used to considerable advantage in most instances, it should be incorporated into any developmentally oriented test station. If desired, it can be used in conjunction with a low-speed system to achieve a more optimal utilization of equipment.

2.4.3 Entering

Before the test data can be digitally sampled some analog filtering must be accomplished to avoid aliasing errors. The Nyquist sampling theorem states that the data sampling rate must be at least twice the highest frequency of the signal content if the signal is to be recovered without error. The continuous analog signals are sampled at a fixed interval, A , and converted into digital signals which are then used for digital calculations. The sampled signal can be represented by a multiplication of the original signal, s ( t ) , and a sequence of impulse functions, i f t ) .

Sj(t) = s( t ) i ( t ) . (2.12)

Using the theorem of convolution. Equation (2.12) is written in the frequency domain as:

Sj(0 = J°° S(f -g) I (g)dg . (2.13) — oo

where 1(g) is the transform of i(t) . The spectrum of the sampler output can be expressed as follows:

S|(0 = l ± S(f-J). (2.14) n = - o o N / n

Equation (2.14) shows that the sampled signal Sj(t) has a transform with period 1/A . If the sampling interval is such that S(f) falls to zero before |f| = 1/2A , then Sj(f) is simply a periodic version of S(f) as shown in Figure 2.10. This means that it is possible to recover S(f) from Sj(f) by multiplying Sj(f) by H(f ) , where

H(f) = 1 |f| < — 2A

H(f) = 0 |fl > -k :A

(2.15)

If, on the other hand, S(f) is not zero above |f| = 1/2A , the frequency components above 1/(2A) in S(f) appear in Sj(f) as shown in Figure 2.11. The frequency fn = 1/(2A) , Nyquist rate, is sometimes referred to as the folding frequency because it is the frequency at which the spectrum Sj(f) is folded back on itself (aliasing).

After the data has been sampled it can be further filtered digitally and then resampled at a lower rate before it is finally stored. The main elements of this data processing sequence can be represented by Figure 2.12. The analog filter bandlimits the raw analog signal so that the aliasing error at the Nyquist frequency is kept within tolerable limits. The sharper the filter cut off can be made, the slower the sampling rate can be for the same amount of aliasing error. After sampling the filtered analog signal, digital filters are used to provide an ideal low-pass filter. The frequency characteristics of the ideal low-pass digital filter are:

19

Fig.2.10 Transform of signal whose frequency content is less than 1/(2A)

2 A

- 3 2A

- 1 A

- 1

7 T 0 1

2A~ 1 A

3 -2

Fig.2.11 Transform of signal whose frequency content is greater than 1/(2A)

F(f) = 1

= 0

- B < f < B

elsewhere , (2.16)

where B is the cutoff frequency. The degree of digital filtering performed is determined by the ratio of one-half the initial sampling rate 1/2A to the digital low-pass filter bandwidth B . After digital filtering the data are resampled at a lower rate for further analysis purposes. The decimation ratio, k , is the ratio of original samples to new samples. Although it is only necessary to resample at a frequency of 2/B to satisfy the Nyquist criteria, a resampling rate at least six times higher than the filter cutoff frequency is desirable to create meaningful graphical displays. In other words, six or more points per cycle are needed to display a sinewave of the highest sampled frequency.

2.4.4 Data Handling

Before the test data can be analyzed and displayed in its final reduced form it must be handled in many ways by the computer. This section discusses the essential data manipulation operations and some of the checks which can be performed to insure that reliable results are achieved.

ACQUISITION: The first data handling operation, that of data acquisition, is performed by the computer using a high-speed data acquisition system, a low-speed data acquisition system and the inputs from several other digital output devices. Data acquisition samples can be initiated on the basis of (i) elapsed time, (ii) elapsed table rotation angle, or (iii) elapsed time increments after the table has reached a given angle and allowed to settle. Any combina­tion of these data sampling modes can be selected by the test operator using the computer test command sequence builder task. As often occurs in practice, the information to be monitored is made up of signals which vary at widely different rates. Therefore the sampling rates can be very different for each signal. Many data compression and redundancy reduction schemes have been proposed to reduce the amount of data which must be stored for a given test sequence. Most of these schemes are based on the recording of a signal and an associated time only when a significant change has occurred in that variable. A matching recovery scheme must also be used for data analysis and presentation. Data compression is not always required however, and the formatting and recovery routines that can be used are greatly simplified. When a digital magnetic tape is used to directly store the data records, space is hardly a problem. In this case each record can contain the entire complement of monitored variables. A new sample is triggered whenever the fastest sampling device requires it. For example, whenever a gyro test is underway, the high speed data acquisition system will trigger another sample every one to ten seconds as required. In between gyro tests the low-speed data acquisition system (LSDAS) will trigger a sample of the environmental parameters every one to ten minutes as selected by the operator. If for some reason the LSDAS is inactive, the gyrocompass system will trigger a sample whenever its four-position cycle is completed, approximately every twelve minutes.

20

I

I

\

RAW ANALOG SIGNAL

ANALOG FILTER

FILTERED ANALOG SIGNAL

SAMPLER

SAMPLED DATA

DIGITAL FILTER

DIGITALLY FILTERED DATA

SAMPLER

SAMPLED, FILTERED DATA

Fig.2.12 Data processing sequence

All samples are put in the same data block format and recorded on tape. This data block consists of 64 high-speed gyro test data channels, 32 low-speed environmental data channels, 16 a.c. and d.c. power supply monitor channels, 2 table angles, a gyrocompass azimuth channel, 26 channels for gyro excitation level settings, and the test times at which the various devices were sampled.

LIMITS; Before the data block is actually written to tape several checks are made. First of all, the measured values are compared with preset limits. If a particular channel is found to exceed either the high or low limit an alarm message is written on the operator's visual display terminal. This immediate notification that a threshold has been exceeded allows the test operator to take some remedial action before the test is invalidated. In those few cases where the out-of-limit condition can be catostrophic to the instrument under test, the computer is programmed to take automatic action to turn off the wheel supply or temperature controller as the case may dictate. These automatic shutdowns are rarely activated, since it is only a fail safe operation in the event that the analog monitor and automatic shutdown unit malfunctions.

FORMATTING: After all of the test data in a given sample has been checked, it is converted to the desired format for tape recording. If the test operator so desires, he may set up a set of transformation operations that will be performed on each data record. Mathematical operations such as addition, subtraction, multiplication or division by another channel or by constants can be performed. This permits bias or scale factor corrections to be completed before recording. The sum, difference or product of two variables can also be calculated and stored. This finalized data block is then used by a data display task that writes a set of 15 fixed and 15 selectable channels to the visual display terminal during actual test operations. This display gives the operator a single consolidated place where he can observe the status of the entire test as it progresses.

RECORDING: Two magnetic tape decks are used for the recording operation for several reasons. The recording is automatically switched from one tape deck to the other if (i) a malfunction occurs, (ii) the tape reel becomes full, or (iii) the operator desires to read the test data recorded up until that time. In this last case, that data is read directly into core where it can be used for analysis and/or display.

21

EDITING; After the test data has been read into core or onto a disc file by a background operating system subroutine, it is generally plotted on the graphic display terminal. This allows the experimenter to get a quick look at the recorded signals and to edit any extraneous points required. The editing is accomplished using a joystick hand controller. This device is used to position a set of crosshairs on the desired point to be inserted. The extrane­ous point which lies on the vertical line will be replaced by the selected point when the letter R is struck on the keyboard. After all extraneous points in a channel have been edited, the entire channel is rewritten to the disc data matrix.

TRANSFORMATION; Following manual editing the operator is asked to enter any desired transformation parameters. This is a more comprehensive set of mathematical operations than is permitted during the real-time operations. Additional functions, such as integration, differentiations, time delay, exponential, sine, cosine, arcsine, arccosine, log, powers and coordinate transformations can be selected. After these operations have been performed by the computer, a low-pass digital filter can be applied to each of the channels. This routine takes care of automatically editing the data and filling in for any missing points. The filter output can then be resampled at any slower integer rate.

PLOTTING: The mean, standard deviation, maximum and minimum values for each data channel are calculated and written on a new tape file along with the filtered data. A short and a long title for each channel are also written at the beginning of the new file. From this final conditioned data tape the operator can select any or all channels to be plotted, scaled and labeled on the graphic display terminal. A hard copy is then made for later examination or report presentation. The conditioned data can be read and transmitted via a modem interface to a Burroughs B6700 data processing computer. Here detailed statistical analysis can be performed. The reduced data is then transmitted back to the terminal where it is either plotted, printed or stored on tape. All of the above mentioned operations are commanded from one of the two display terminals. Interactive programs are used to query the operator and to prompt acceptable responses in each case.

2.4.5 Displays

Numerous displays are used throughout the test electronics and the computer handling routines to monitor the status of the test operations. These displays have been categorized into analog and digital displays for the sake of presentation only. It has been found from experience that a variety of displays along the data processing route is essential if any logical sense is to be made out of the complex of test variables and manipulation steps.

ANALOG: The easiest method for examining analog signals is through the use of panel meters. These have been included on the temperature monitor and alarm, the float position monitor, the analog torque loop, the temperature controllers, temperature measurement electronics, line voltage regulators and differential wattmeter. A quick look indication of proper gyro and test electronics operation can be obtained from these visual display meters. In addition, many indicator lights are used to display the status or mode of a certain device. Audio and visual alarms are incorporated into those units that perform a critical monitor function, such as measuring the average gyro temperature or monitoring the air pressure on the test table. A phase angle voltmeter, a multimeter and an oscilloscope are included in the test electronics console so that the many front panel test points can be examined and controls can be adjusted. An eight channel strip chart recorder is also included in the monitor and display equipment so that time related variables can be observed without the requirement for data acquisition or computer processing. In spite of the sophisticated computer data handling routines, this standard device proves to be quite valuable during initial checkout and calibration of signal monitor electronics. The changes that occur in such parameters as the float positions or temperature gradients can be readily observed as external stimuli arc applied to the gyro. A sixty-four channel signal selector panel has been incorporated into the electronics console so that any eight of these signals can be easily routed to any of the eight analog recorder channels.

DIGITAL: The digital displays are primarily concentrated on the Alphanumeric Visual display terminal. On this device any of the 145 recorded data channels can be displayed in real time. Other preselected information such as the channel limits, transformation parameters, table angles, data acquisition parameters, or gyro electronic programming sequences can also be reviewed on this screen. The table readout angle is separately displayed on the table electronics console along with the primary and secondary axis rates, dwell time and cycle number. A digital readout clock is used to display the total time into a test in days, hours, minutes and seconds. The computer interface unit for the gyro electronics contains a bank of digital display lights that can be read and interpreted under either manual or computer control. The four programmable low pass amplifiers also contain lights and switches that perform essentially the same functions. The other primary computer output display devices which have already been mentioned are the graphic display terminal, the line printer and the ASR 33 teletype that keeps a constant log of the computer's activities, including the on and off times for individual tasks. All of these digital display devices are designed to keep the test operator fully informed of the status of any gyro test.

22

2.5 Evaluation

2.5.1 Overview

The philosophy to be used in the specification of a research test facility is to develop a general purpose data gathering, editing, and analysis capability to aid in the understanding and interpretation of error phenomena associated with the performance of precision inertial guidance components. The requirement for on-line data analysis and display stems from the desire to do research investigations on many variables and to compare various test techniques and analysis methods. The time required to obtain results from a computation center can be intolerable when the characteristics of the environment and inertial sensors may be constantly changing. Interpretation of the test data and analysis of intermediate results while the test is still in progress allows the researcher to make controlled changes and immediately observe the effects of these changes. Real-time interaction with the test in this fashion can greatly increase the number of hypotheses that can be verified during a particular test sequence. Various test procedures can be compared and different digital filtering and/or analysis techniques can be implemented to determine the best methods for a particular sensor and test configuration. Generalized results obtained from these investigations can be used to make knowledgeable decisions concerning the test and evaluations of other instruments to meet certain test requirements. Flexibility in the use of a research-oriented laboratory digital system is the key to its effectiveness. Unforeseen requirements will certainly develop as the experiments are changed to deepen our understanding of a particular error mechanism and to isolate various cause and effect relationships.

A comprehensive package of Software to Automate Guidance Experiments (STAGE) has been developed to assist the test operator and researcher in conducting a great variety of gyro tests. The design goals used in the development of this software operating system included:

(a) Operator simplicity

(b) Test design flexibility

(c) Expandability

(d) Interactive capability

(e) Real-time monitor

( 0 Modularity.

These factors are desirable in a research test environment inasmuch as they allow the principal investigator to implement various test methods and to quickly ascertain the results of his inputs in order to iterate on an optimum procedure.

Operator simplicity relieves the researcher of the burden of obtaining a thorough understanding of the computers' job control language, file maintenance and utility function. Anyone can quickly learn to set up and run a particular test from the pre-programmed output instructions which are printed on the Alphanumeric Display Terminal. Computer checks and defaults are used to prevent operator errors or oversights from interfering with proper test operations. In spite of its inherently simple operation the software must be designed with a maximum degree of flexibility so that new procedures and test techniques can be incorporated into its operation. This is mainly accomplished through the use of a modular software concept. Short subroutines are linked together in many different ways to complete the software tasks. These subroutines can be easily modified, replaced or eliminated as the need arises for various changes to be made in the software environment. This technique also creates an almost unlimited expansion capability. New subroutines are written and the executive module is modified to include its operation as another option to the experimenter. The only restrictions which do occur are imposed by core limitations and the speed at which all tasks can be completed and still maintain the real-time operation. Since most of the tests which are performed on a gyroscope are relatively slow compared with the computer's operating speed, this does not usually present any serious problems. Operator interaction with the test is made possible by the extensive use of visual displays and a graphics terminal. The capability to modify the test parameters or to edit the test data through the use of an interactive terminal gives the experimenter considerably more time to reflect upon the actual test results since less of his time is spent in mere data manipulation.

One of the most important functions inherent in the research requirement is the ability to handle multiple tasks simultaneously. For example, while stored data is being analyzed for display and interpretation, a priority interrupt to take another data sample may occur. This peripheral device must be serviced by the computer and then the desired digital filtering, calculations and storage requirements must be performed. In addition, the computer must monitor environmental parameters, issue control commands, service peripheral devices, and allow program compilation, debugging and copying at the same time. These requirements demand the use of a fully integrated hardware-software system operating in a multilevel programming environment, consisting of foreground, middleground and background processing for optimum response to real-time events while concurrently handling data processing jobs.

In the foreground, the highspeed interrupt facilities provide optimum response to real-time events. Foreground programs, such as those used for controlling highspeed sampling of measurements and data transfers to and from

23

peripherals, are connected directly to dedicated hardware interrupt levels. The hardware interrupt facilities automatically resolve all priority and scheduling conflicts and thus eliminate delays between requests and execution of foreground programs.

The middleground provides a real-time multiprogramming environment for scheduling tasks on the basis of elapsed time, time of day, requests by other tasks, operator request, and even by hardware interrupts. The execution of all middleground tasks is governed by a software priority structure that is entirely separate from the hardware interrupt system. Thus, as many middleground tasks as needed to handle the workload can be dynamically programmed. This provides the ideal environment for multi-user, timesharing requirements because of its low overhead.

The background provides the ideal environment for routine, nonreal-time jobs like program assembly, Fortran IV compilation, debugging, interfile copying, file maintenance, and source editing. Background processing runs on a time-available basis concurrent with foreground and middleground processing. Because the background is serviced only after all tasks critical to real-time activities have been processed, it exploits the full resources of the system without jeopardizing the system's real-time capabilities. Data analysis, formatting and display can be performed in near-real time using the Background Operating System (BOS). This allows the experimenter to obtain intermediate and quick-look test results while the actual test is still underway. A decision can thus be made to extend, modify or terminate a particular test as indicated by the nature of the current test data.

2.5.2 Computer Hardware

Several digital computers have been incorporated into the total gyro test laboratory operation. One (HP 21 14B) is dedicated to the on-line digital control of the seismic isolation system. Another (SYSTEMS 81 OB) is used for the real-time control, monitor and intermediate analysis of the gyro testing operations. Still a third (BURROUGHS B6700) is connected to the laboratory computer via a data link. This machine is used to carry out the large data analysis routines with sufficient precision and speed. Its results are transmitted back to the test laboratory for storage, listing and/or display on the standard output devices. The heart of the computer operation lies in the 81 OB digital controller. It must be capable of functioning continuously throughout a gyro test, activating tasks and commanding peripheral devices that are required by the test. A description of the hardware which has been selected to perform this function will be given in this section. A description of the software which has been developed to carry out the particular research tasks will be given in the next section.

A great variety of extremely powerful, yet relatively low cost mini-computer systems have proliferated the computer market in recent years. Several of these systems would be adequate to carry out the required tasks. The particular hardware which was selected by the F.J.Seller Research Laboratory was based on numerous considerations such as cost, reliability, supported software, programming ease and interface simplicity, but not necessarily in that order. With the rapid pace being set by the leading computer development manufacturers, I am certain that the present equipment will be outmoded in a few years. Consequently, it will be necessary to separate the desired test capability into its basic elements and to review current production hardware and software before the best system for the job can be assessed. However, it is important to realize that the amount of resources invested in peculiar soft­ware development will eventually exceed the cost of the original hardware. It therefore becomes very difficult to upgrade a machine at a later time unless the system manufacturer has made his new equipment hardware and software compatible with existing equipment. More often than not, a large portion of the software must be redone whenever this occurs.

The computer related hardware which is currently used by the FJSRL in its Inertial Guidance Research Division is as follows:

Central Processing Unit. 32,000 words of 16-bit core memory: 96 hardware-wired priority interrupt levels, 750 nano second cycle time; 67 programmable instruction set; resident Real Time Executive (RTX) software; two direct block transfer control units (BTC); automatic restart; stall alarm.

Moving Head Disc. 1.5 million words of 16-bit storage; 60 milliseconds average track access time; directory; supported software.

Magnetic Tape Unit. Two seven track tape drives operate from one control unit; 200, 556, or 800 bits per inch packing density, 45 inches per second slew rate; 10 inch tape reels.

Digital Input/Output Unit. 16 input and 16 output channels, 16-bits each; 4 asynchronous communication channels.

Card Reader. 600 cards per minute; block transfer controlled.

Line Printer. 500 lines per minute; 132 columns; 48 lines per 8'/2 x 11 inch page; 7 x 1 0 matrix letters; silent operation; electrostatic printing.

24

High Speed Data Acquisition System. 64 channels; 14 bits plus sign resolution; V. LSB accuracy, 100,000 conversions per second sampling rate: random, sequential or block scan.

Low Speed Data Acquisition System. 100 double-ended input channels; d.c., a.c., or ohms input mode; 10 parts per million resolution; up to 50 samples per second depending on mode selected.

Graphic Display Terminal. Alphanumeric character generator; storage tube display; hard copy attachment; 0.01 inch graphics resolution, 6.1 x 8.3 inch screen; vector, point plot and incremental plot modes; 94 printing characters; joystick controller; refresh scratch pad display line; plot subroutine package.

Alphanumeric Display Terminal. 64 ASCII character set on keyboard; foreground/background text capability; roll-up feature: remote send to printer; full edit; addressable cursor.

Teleprinter. ASR 33; paper tape reader and punch; 10 characters per second print, punch and read.

Gyro Electronics Interface. 16 parallel or serial 16-bit data channels; manual overide; addressable; read/write capability.

Digital Clock. 1 millisecond resolution; 12 day storage; direct access by computer.

Push Button Interrupts. 12 button; tied to hardware interrupt registers; activate selected software tasks.

Table Data Display Terminal. Two axis rate display; cycle number display; dwell time display; 2 digital to analog converters; 2 encoders and clock time buffers.

2.5.3 Stage

From the point of view of an RTX based software system it is advantageous to segregate the STAGE package into two major parts. The two parts include test operations and analysis operations. The test operations include the real-time, closed-loop control of the test equipment, facilities for scheduled and emergency redirection of the test by the test director and the collection of data for evaluation of the results of the test. The test operations are time sensitive and therefore require priority handling by the system. Subsets of the test operations functions may be operated for purposes of subsystem testing, calibration, etc.

The analysis operations include the retrieval of collected data, data filtering and transformation, statistical analyses and report generation. The analysis operations may be performed in the background since delays caused by priority interrupts cannot jeopardize the test. The test execution and analysis operations interface only through the data base. The data base is created by the test operations and processed by the analysis operations. The data base includes the information obtained from the test fixtures, the environmental sensors, test control settings, digital clocks, and the data acquisition systems.

The Systems 810B Real-Time Executive (RTX) is a modular software system for controlling, scheduling, and monitoring the activities of programs in a real-time multiprogramming environment. It also has provision for the development and execution of programs in a background mode. The full capabilities are implemented through a combination of hardware interrupt control and software task scheduling.

RTX schedules all programs other than those that are triggered by hardware interrupts. RTX is capable of initiating tasks on a time-of-day, elapsed time, immediate, interrupt, round-robin or operator-requested priority basis. Tasks may initiate other tasks. RTX dynamically allocates and protects core storage, loads non-resident programs and provides a log of its activities.

The Input/Output Control module of RTX processes all I/O functions including those of the Real-Time and Batch Operating Systems. A dispatcher identifies each request and schedules the appropriate I/O handler to process the request. The handler performs any necessary formating or editing, connects the interrupt driven software to the hardware interrupt and, upon completion of the transfer, evaluates the success of the I/O operation.

Interrupt processors, also called I/O drivers, operate in the foreground and are triggered by hardware interrupts. This feature minimizes the overhead associated with interrupt processing, provides a wired priority scheme and auto­matic preemption of any middleground or background task. The selection of the proper priority level for each of the system interrupts is an important systems design consideration. The major modules of the standard software are:

(a) Multiprogramming Executive

(b) Executive Services

(c) Input/Output Control Scheduler

(d) Input/Output Handlers (for standard peripherals)

(e) Interrupt Processors (for standard peripherals)

25

(0 Executive Subroutines

(g) Loaders

(h) System Generation

(i) Batch Operating System, including utility, debug, and foreground load module builder, FORTRAN compiler. Macro Assembler and Libraries.

Test Operations: The test operations software contains two main RTX tasks. They are the Test Director Communications Task and the Test Execution Task. Each task consists of a control program and several processing modules. These tasks are designed to facilitate the addition, deletion or modification of modules to meet changing requirements. A diagram which depicts the interconnection of tasks and handlers is shown in Figure 2.13.

S T A G E

TEST PREPARATION

Q LL

c

*<

CONTROL WORD 12

TEST EXECUTION

— CALL

— SAVE

— DISPLAY

— RECORD

— TERMINATE

— WA IT

— EXECUTE

FECKER HANDLERS-CONTROL

WORD 9

3

to

a zr. < O Ol-*,

FECKER

DATA ACQUISITION

GYRO ELECTRONICS

DISPLAY

ALARMS 8 LIMITS

TRANSFORMATION

SPARE

DUMP ROUTINE

GYRO ELECTRONICS HANDLER -*

I

HIGH SPEED DAS HANDLER - < I

LOW SPEED DAS HANDLER

ALS HANDLER

•>• LIMITS CHECK HANDLER

••TRANSFORMATION HANDLER

I—•* DISPLAY HANDLER -*

SYSTEM TAPE HANDLER-

POWER FAILURE 8 MONITOR

RESTART

BOS ON

'TAPE 8 CARD READER ON

TEST OPCOM ON

PANIC STOP

'SHUTDOWN

.2 SEC

WAIT

Fig.2.13 Flow diagram of test operations software

The Test Director Communications task consists of a control program which decodes and stores the data typed into the Test Director's Console. This task reserves a block of disc space for the storage of test parameters. Several different tests can be entered, edited, listed, stored, recalled, and executed at any time through the test operator's communication console (ADT). An interactive computer program using the foreground/background feature of this terminal can be initiated by pushing a single hardware interrupt button. Detailed operator instructions, preprinted forms and relevant questions are presented on the screen to prompt the correct operator responses. Input parameters

are checked for validity and are rejected whenever an illegal entry is detected. Complete text editing is allowed using the tab and cursor positioning keys before the information is transmitted to the computer. Thus, an untrained operator can quickly learn to set up the many parameters required to execute a novel test and can obtain an instant, centralized display of all pertinent parameters which are undergoing a change during the test execution.

Analytical Operations: The Analysis Operations Software provides the capability to retrieve, manipulate and analyze the test data. The test data is recorded by the Test Execution task on magnetic tape in real time. The Analysis Operation Software is operated in the background under control of the Batch Operating System. It consists of a data retrieval module, transformation module, digital filter module, statistics module and a sequential plotting module.

The data retrieval module reads the data from magnetic tape into a numerical array in core. This array can be accessed by either FORTRAN, BASIC or Assembly language programs. A set of transformation parameters is entered from the keyboard through the scratch pad area of the Graphics Display Terminal. These parameters set up mathematical subroutines which will add, subtract, multiply, divide, integrate, differentiate, log, or delay the selected channels of data after they are read into the storage array. This allows the operator to form a data block which has the proper sequence, scale factor, bias, etc., for use in later data analysis routines.

A digital filter module can be summoned to filter all channels to eliminate unwanted sampling noise. The preliminary statistics of each variable, including its mean, standard deviation, maximum and minimum values, are calculated and stored in order to facilitate a simplified plotting routine. Each data channel is then plotted on the Graphic Display Terminal as selected by the test operator. Data editing can be performed at this point by positioning a set of crosshairs which are controlled by a joystick over the point to be edited. The computer will respond by accepting a keyboard entered correction or by calculating an interpolated value at the operator's option. The edited data may then be stored for later use.

Data analysis routines are available on a Burrough's B6700 remotely tied computer to perform the following: Regression, Correlation, Probability Density Function, Power Spectral Density Analysis, Multiple Time Series Analysis, Central Tendency and other commonly used statistical routines. A conversational language is used to transmit the data, set up the desired program parameters, call library routines and to return the numerical results. These can then be directed to the Graphics Terminal, Disc, Magnetic Tape or to the Line Printer for later interpretation.

3. GYRO TESTING TECHNIQUES

3.1 Two-Axis Tumble Test

3.1.1 Introduction

The standard torque-to-balance test19, which is most often used to determine the coefficients of the static gyro­scope error model in a one-g specific force environment, has many inherent limitations. Most noteworthy is the fact that the gyro must be repositioned (from SA up to SA down or from OA up to OA down) after each rotation of the test table. This not only introduces a transient into the instrument, especially if it is accomplished manually, but also necessitates the combining of data from various orientations to calculate a single solution for the drift coefficients. These test restrictions have been imposed upon most test facilities due to the limitations in most gyro rate tables. If the axis of rotation about which the gyro is normally reoriented is automated along with the standard polar axis drive, then numerous possibilities for positioning the gyro with respect to the earth rate and the gravity vector are available. Theoretically, only nine gyro positions are necessary to uniquely solve for nine gyro drift coefficients. However, due to the advantages accrued from taking redundant data, a two axis tumble test has been devised in which a new set of coefficients can be calculated, although correlated with past solutions, after each new position is completed. This method is considerably faster than previous gyro test methods; thus, it provides more accurate information on the short-term instabilities of the gyro drift coefficients. In this section, a description of the two-axis tumble test procedure and data analysis will be given.

3.1.2 Gyroscope Performance Model

In the ideal case, the gyro's output signal can be represented by a sensitivity to angular input rates.

SJQI = co\ , (3.1)

where SJQ is the torque generator scale factor, i is the torque-to-balance current, and co\ is the angular input rate of the gyro case. In practice, however, there are a number of error torques acting on the float which must be counteracted by the torque generator current.

Using a linear expansion of these error sources, we can write the following:

11

Srci = Wd = co, + Z ( ^ ) 6 x i 0-2)

where 9Wd/9Xj is called an error coefficient or drift sensitivity.

These error coefficients are usually described by a second-order expansion of the specific force inputs along each of the three orthogonal gyro axes (see Figure 3.1).

O U T P U T

A X I S ,.

I N P U T ^ . . S P I N A X I S * A X I S

Fig.3.1 Axes of a single-degree-of-freedom gyro

The mathematic error model is then written as

Wd = co, + D F + D,f* + Dsf0 + Dsfs + D,,ff + D00f20 + D s sf | + D,0f,f0 + D o s f 0 f s + D,sf,fs + 5co (3.3)

where

Wd = total gyro drift rate

Dp = that component of gyro drift which is insensitive to specific force inputs

D; = sensitivity of gyro drift rate to specific force along the gyro j axis

DjK = sensitivity of gyro drift rate to the product of specific forces along the gyro j and k axes

fj = specific force directed along the gyro j axis

6to = gyro uncertainty consisting of that drift rate which is unaccounted for by the other terms in the error model.

3.1.3 Test Description

A test fixture which is capable of simultaneously positioning the gyro about two orthogonal axes is essential to this test. The principal fixture axis is parallel to the earth rate vector while the secondary fixture axis is perpendicular to this axis and rotates with the primary axis. The gyro is mounted so that its input axis is parallel to the fixture secondary axis (see Figure 3.2). Because this axis always remains perpendicular to earth's polar axis, the gyro's input axis senses no earth rate except for small misalignment angles. The starting position can be arbitrarily chosen with the gyro input axis pointing west while the spin axis lies parallel to minus earth axis.

Since a continuous rotational rate of the fixture secondary axis would be directly sensed by the gyro input axis, a pause type tumbling test is normally performed. This eliminates data sampling while the gyro is undergoing a large input rate and eliminates the uncertainty due to table rate drive inaccuracies. However, the gyro must be allowed to settle after each new position is reached before the torquer current can be sampled. Several repeated samples can be taken at each position if a weighted solution is to be calculated or if a smoothed average is desired.

This testing method eliminates the requirement for counter-rotations. Also, there is no need to stop the rotation once the sequence required for one solution is completed. Each additional data point, when combined with the previous data provides a new, although correlated, solution for the drift coefficient. Recursive least squares methods or Kalman filtering solutions can be applied to this test data as desired.

3.1.4 Solution for Drift Coefficients

In order to obtain a solution for the drift coefficients given in Equation (3.3), an appropriate ratio of angular motion between the primary and secondary axes must first be selected. The amount of correlation that results between the various coordinate functions depends directly on this ratio. Letting the ratio of angular positions about the two table axes be:

28

Fig.3.2 Alignment of gyro and test table for the two-axis tumble test

mO

nfl" in

n

where

md is the rotation angle about the principal table axis which is parallel to the earth's polar axis.

nd is the rotation angle about the secondary table axis which is parallel to the gyro input axis.

The components of specific forces which lie along the gyro input, output and spin axes are expressed in terms of 6 as follows:

f| = — g cos L sin md

fo = — g cos L cos md cos nd — g sin L sin nd y

f$ = g cos L cos md sin n(9 — g sin L cos nd

where g is the specific force due to gravity and L is the astronomic latitude angle at the test table locations

Substituting the expressions given in Equation (3.4) into Equation (3.3) and rearranging terms gives:

Wd = to, + D F + j D „ cos2L + J - D Q O ' W L + sin2L) + J-Dss(cos2L + 2 sin2L) +

+ \ D o s sin L cos L cos (m — 2n)0 -I- ^ ( D s s — D Q Q ) s i n L cos L sin (m — 2n)0 +

+ (— ^ D Q cos L + j D , 0 sin L cos L) cos (m — n)d + (— *Dg cos L +

+ \ D,s sin L cos L) sin (m — n)d — D§ sin L cos n(? — D Q sin L sin nd +

+ | ( D o o ~ Dss) cos2L cos 2(m — n)0 + gDgs cos2L sin 2(m — n)0 — Dj cos L sin md +

+ ( D s s - D 0 0 ) ( i sin2L - \ cos2L) cos 2nd + D o s ( l sin2L - \ cos2L) sin 2n0 -

(3.4)

>.(3.5) cont'd

: "

— \ D , S c o s 2 L c o s ( 2 m - n ) 0 + \ D i 0 cos2L sin (2m - n)0 + ( - ^ D Q COS L -

— \ DJO sin L cos L) COS (m + n)0 + (^Dg sin L + ^D,s sin L cos L) sin (m + n)0 +

+ (^D, cos2L + J-DQQ cos2L + \ D S S cos2L) cos 2m0 + ^ D o s s i n Lcos Lcos(m + 2n)0 +

+ i(°00 - DSS) s i n L c o s L s i n ( m + 2 n ) e + i D IS c o s 2 L c o s <2 m + n)*9 +

+ i D , 0 cos2L sin (2m + n)0 + i ( D 0 o _ D s s ) cos2L cos 2(m + n)0 -

— gDos cos2L sin 2(m + n)0 .

>(3.5) cont'd

The values chosen for m and n determine the degree of correlation which will exist among the coordinate functions. However the coordinate function for Dp, D„ , DQQ > a n d Dgg will always show some correlation since they all contain bias components. Correlation is also dependent upon the number of data points used in the solution of the coefficients. This is true because sin a0 is correlated with sin b0 , and cos a0 is correlated with cos b0 when |a ± b| equals the number of equispaced data points used. Computer simulations studies20 have shown that low correlation exists for ratios such as 5:2, 5:3, 5:4, 6 :1 , 6:5, 7:2, 7:3, 7:4, 7:6, etc. Other ratios, such as 4:3 and 7:5 do not permit coefficient separation due to high correlation.

A solution for nine gyro drift coefficients (omitting D Q Q ) can be obtained by the method of least squares. If the error in the test data is normally distributed and uncorrelated with zero mean, the least squares solution yields the maximum likelihood, minimum variance estimate of the drift coefficients. By defining Y to be the vector of gyro output observation, X to be the matrix of coordinate functions, D to be the vector of drift coefficients to be estimated, and e to be the vector of residual errors, the assumed gyro error model is written as:

Y = X D + e (3.6)

The criteria used to obtain the best estimates of the model coefficients, D , is to minimize the sum of squares of the residuals.

S e? = e2 + e2 + ... e2 —T — = e ' e (3.7)

From Equation (3.6):

= Y - X D

T -e ' e =

(Y - XD) ' (Y - XD)

e T e = YTY D T X T Y YTXD + D T X T XD

(3.8)

Since all terms in this equation are scalars,

YTXD = (YTXD)T = DTXTY (3.9)

and FTe = Y T Y - 2DTXTY + D T X T XD . (3.10)

Minimizing this quantity, e T e , is done by taking the first derivative with respect to the coefficients to be estimated, D , and setting the result equal to zero.

3 e T e

3D

XTXD

: X ' Y + : X ' X D = o

XTY. (3.11)

Solving for D , we obtain

D = (X T X)- 'X T Y (3.12)

where (XTX) ' is referred to as the inverse least squares matrix.

The familiar statistical parameters which are associated with a least-squares regression analysis are obtained from the analysis of variance. From the gyro output signal measurement (dependent variable), the total sum of squares is calculated, YTY . This must be equal to the sum of squares due to the regression D T X T Y , and the sum of squares of the residuals, (YTY — DTXTY). One test for the goodness-of-fit of the proposed model is the square of the multiple correlation coefficient, R .

30

«• - -$? . The quantity R2 represents that fraction of the total sum of squares which can be accounted for by the

regression model. It is often expressed as a percentage by multiplying by 100.

If the error model is accurate and complete and the coefficients are constant, then the standard error of the test, a , can be estimated from the mean square value of the residuals.

1 HZ e? (3.14)

N - q N=l '

where (N — q) is the number of degrees of freedom, q is taken as the number of significant terms in the error model.

The standard error of a particular drift coefficient, D; , is calculated as

a i = a(A j j)1 / 2 * ( 3 ' 5 )

where Ajj is the term on the jth row and the jth column of (XTX)~' , the inverse least squares matrix. The quantity a-, is an estimate of the standard deviation of the error in a drift coefficient if the mean of the error is assumed to be zero. It is a useful statistic in that it permits a T-test to be performed. A test of the hypothesis that the estimated value of a coefficient in the least squares regression model is significantly different from zero can be checked by comparing the coefficient with its standard error.

D i tj = - J - (3.16)

where tj (n — q, C) is the (1 — C/2) percentage point of a t distribution with (n — q) degrees of freedom (the number of degrees of freedom on which the estimate o2 is based). If we assume that the variations in the measure­ment of Dj are normally distributed, then we can assign 100(1 — C)% confidence limits for D, by calculating:

Dj ± t - O i - q , 1 -C /2 )0 . j . (3.17)

For example, with 1000 degrees of freedom, the 99% confidence limit value obtained for 1,(1000, 0.01) from standard statistical tables is 2.58. We could then state with 99% confidence that the true value of D, lies in the interval

D, - 2.580j < D, < D, + 2.58OJ . (3.18)

A test for the significance of the overall regression equation is obtained by comparing the sum of squares due to regression with the estimated mean-square residual.

D T X T Y G = \ — . (3.19)

a2

G follows a Snedecor's F distribution with q degrees of freedom for the greater mean-square and n — q degrees of freedom for the lesser mean-square. For a given confidence level, C , the fact that the observed mean-square ratio, G , exceeds F(q — I, n — q,C) means that a "statistically significant" regression has been obtained. In other words, the proportion of the variation observed in the data, which has been accounted for by the error model, is greater than would be expected by chance in 100(C)% of similar sets of data with the same values of n and X .

3.1.5 Summary

This section has presented one possible mechanization and analysis procedure for a two axis tumble test. Many other variations of this scheme are possible. An optimum arrangement of table positions to minimize the uncertainty in the drift coefficient estimates has yet to be determined.

The procedure presented does permit a much more rapid determination of the gyro drift coefficients than previously used techniques. A sliding window or Kalman filter solution can be applied to the test data to obtain a continuous solution of the individual coefficient variation. Thus, an improved measure of the gyro's short-term instabilities can be made. A combination of this relatively simple test sequencing with variations in other test parameters which influence the gyro's output response, will allow a solution for the coefficients of an extended gyro error model. This concept will be discussed in the following section.

31

3.2 Gyro Error Modeling

3.2.1 Introduction

The present trend toward a more complex statistical description of gyro uncertainty leads to an increasing requirement for more test data which are extremely costly to obtain and cover only a small segment of the operating environment of a gyro. There is still much room for improvement in the gyro performance through a detailed investigation of the deterministic error mechanisms working within the instrument. Once these have been isolated and mathematically modeled, the statistical modeling of the remaining uncertainty can more properly proceed.

/Vlthough the exact mathematical relationship which describes the greater portion of gyro drift uncertainty is likely to be extremely complex and nonlinear, a linearized model would prove to be of great value as a first-order description over a limited range. Continued research in this area is required to isolate the predominant error sources and to systematically refine the mathematical relationships involved.

Once the causes of gyro drift rate variations are suitably isolated, a means of eliminating or substantially reducing their effects is immediately available through output signal compensation. This means that more reliable and consistent results can be obtained during laboratory testing and that improvement in navigation system performance is possible.

One possible approach is to consider a gyro as a multiple input device having one useful output. The objective is to describe this output as a function of the measured input variables so as to further reduce the uncertainty in the residual. Multivariate statistical analysis techniques are applied to experimental data that are gathered by a pre-described procedure. The object of this analysis is to fit the data to a proposed mathematical model of fixed functional form, estimating values for the undetermined coefficients of that model. Evaluation of several error models is undertaken to identify the most useful prediction of the observed residuals.

The methodology used to identify a suitable gyro error model begins with the development of a mathematical representation of the gyro torque signal (dependent variable) as a function of the disturbance parameters (independent variables). Such a representation can take various forms depending on the degree of complexity desired. Two opposed criteria are to make the equation exact and complete by the inclusion of many terms and, on the other hand, to minimize the number of terms to reduce cost and complexity. The approach taken here is to begin with a very complete and comprehensive model and to then eliminate terms until a useful and yet practical model representation is found. The three considerations used to develop the assumed mathematical model are (i) physical reasoning, (ii) previously observed causal relationships, and (iii) linear expansions to obtain completeness and uniformity. As the model investigation experiment continues, those variables which show no significant effect on the gyro torque signal are eliminated, while a continuing search is made for additional variables which are required to account for systematic trends in the residuals.

Traditionally, the gyro error model has been principally described by a second-order linear expansion of the specific force inputs.

Wd = to, + DF + D,f, + D0f0 + Dsfs + D„ff + Doof2) + D s s f | + D,0f,f0 + D,sf,fs + D o s f 0 f s + 6co (3.20)

where

Wd total gyro drift rate

Dp that component of gyro drift rate which is insensitive to specific force inputs

l)j sensitivity of gyro drift rate to specific force along the gyro j axis

Djj. sensitivity of gyro drift rate to the product of specific force along the gyro j and k axes

fj specific force directed along the gyro j axis

6co gyro uncertainty consisting of that drift rate which is unaccounted for by the other terms in the error model.

From laboratory test experience, it is known that gyro drift rate is sensitive to other variables in addition to specific force. These additional disturbance variables can be broadly classified as float motion, temperature, and power.

To extend the model represented by Equation (3.20) involves a separation of the deterministic effects contained in 6to . These effects can be considered predictable errors rather than unpredictable uncertainties, if the associated sensitivities can be experimentally determined and verified.

3.2.2 Float Motion Errors

The quantity of interest in investigating the effect of float motion on gyro drift rate is the deviation of the float from some nominal position. The displacement ( P S , P | , P Q ) an<l orientation (r<;,r,,ro) of the float with respect

32

to the case is measured by means of the gyro magnetic suspension signals, the translational rates (vs,v, ,v0) and the rotational angular rates (W<5,W,,WQ) can be calculated.

Several previous s tud ies 2 1 , 2 2 , 2 3 have examined the relationship between the motion of the gyro float and the resulting error torques. Generally, it has been shown that these torques are proportional to the float position relative to the case and also to the cross-product of float eccentricity and relative float velocity. Mathematically, this relationship can be expressed in terms of error coefficients as follows:

PSPS + PlPl + POPO + VSVS + VIVI + v ovo + -Vs + Rin + i V o + w s w s + W,w, + W0w0 +

+ NS[PsV| + N, sp,v s + A s , r sw, + A l sr ,w s (3.21)

where the P , V, R, W, N , and A's are sensitivity coefficients to be determined by experimentation.

3.2.3 Temperature Effects

Much research work has gone into the description and control of gyro thermal distribution patterns and error-generating ef fec ts 2 3 , 2 4 , 2 5 . In order to describe this interference parameter in its simplest possible form, the tempera­ture at the six cardinal points of the gyro are monitored. From these measurements the temperature gradients along the input axis (h,), spin axis (h§) and output axis (hG) are determined. Also, by differentiating the temperature time history, the gradient rates of change (q , ,qs ,qo) a r e calculated. In addition to these variables, the average gyro temperature (h^) and the room ambient temperature (IIR) are monitored. The error torques generated by these disturbance variables can be represented as follows:

G|Sh,fs + Gs,hsf, + G 0 0 h 0 f 0 + T G h G + T R h R + B sq s + B*q, + B 0 q 0 + BGqG (3.22)

where the G , T , and B's are error sensitivity coefficients.

3.2.4 Input Power Variations

Internally generated temperature disturbances can result from gyro excitation source power variations. These are mainly wheel power, magnetic suspension power, signal generator primary winding power, torque generator primary winding power, and heater power. Since these thermal disturbances are considerably altered before being sensed by the temperature sensors, they are monitored directly and are included in the error model by the addition of the following terms.

P\VPW + PMSPMS + PSGPSG + pTGPTG + pHPH <3-2 3)

where the P's represent the sensitivity of gyro drift rate to power variations.

3.2.5 Test Methods and Analysis Procedure

A limited amount of test data was taken on a single, representative gyro unit in order to identify instrumentation requirements, to verify test procedures, to implement software techniques and to pave the way for continued research in the areas of gyro error modeling. This testing was designed to show the feasibility of advanced instrument error modeling by developing the fundamental experimental and analytical techniques necessary to satisfy that objective.

In attempting to establish the relationship between all of the variables to be investigated and the gyro output signal, several problems are evident. First, it is impossible to observe the effect of each variable independently of every other variable due to the inherent dependence between variables. Secondly, the large number of disturbance variables being considered dictate that a large volume of data must be accumulated in order to monitor changes which occur in all of these factors. These problems generate the requirement for an analytical procedure which accounts for the correlation between variables.

The basic test procedure employed minimized the correlation problem by controlling the disturbance variables in a mutually orthogonal manner. By forcing the disturbance variables to change according to sine and cosine waves of different frequencies, their time-averaged products tend to vanish. In this way only the causal mechanisms work­ing within the gyro are revealed in the correlation coefficients; the effect of random correlations is minimized or eliminated. Therefore, the observed correlations between variables generated by orthogonal command signals contain the means by which the internal behavior of the gyro can be better understood. The cross-correlation function is used to distinguish the cause from the effect.

Since the key objective of this study is centered around the modeling of gyro error sources, a stepwise multiple linear regression analysis is the essential element of the data analysis. Sufficient flexibility is provided in the computer program to generate new variables from the old and regress on any one or more of these variables. The basic analytical problem to be solved is to determine the best coefficients for the mathematical model, given a record of the gyro's inputs and outputs. This "black box" approach is complicated by the fact that the proposed

33

model may be incomplete or improperly structured. Thus, in a wider sense, the problem is one of model-identification. In this regard, the stepwise regression procedure has been selected in order to gain increased insight into the relative importance of each disturbance variable taken alone or in combination with other variables. The most significant terms are included first and in sequence, until the addition of new variables is of no major consequence in further reducing the residuals. A check of the final residuals is then made to insure that they are representative of a normal, random distribution and do not exhibit any systematic trends.

Once a suitable error model was identified for the particular instrument under test, a tumble test was performed to verify the usefulness of the expanded error model under typical laboratory test conditions. By recording all of the disturbance variables in addition to the gyro error signal during the tumble test, the extended error model coefficients previously determined for this instrument are used to compensate the test data. In this way, the error effects attributable to each disturbance variable are ascertained, in addition to obtaining better estimates for the classical gyro drift coefficients.

3.2.6 Experimental Results

In order to declassify the performance data given in this paper, all of the numbers given in the graphs and tables have been converted to an artificial set of units. The term DRUMS, derived from gyro drift rate unit measurements has been used throughout. Although absolute values are lost in this process, the essential conclusions can still be obtained through a relative comparison of the results.

The parameter variation tests, used to determine the values of the error model coefficients, were conducted using six different gyro orientations (each principal axis up and down). In this way the dependence of each coefficient on input specific force was determined. Figure 3.3 shows the recorded gyro output signal for one of these test runs (spin axis up). The harmonic patterns that are recognizable in this plot were caused by the combined effects of several disturbance variables which were being varied sinusoidally during the test. Analysis of the data revealed that the wheel-supply voltage and current excitations, the torque-generator-supply voltage and current excitations, and the signal-generator-supply voltage and current excitations had little or no effect on the gyro drift rate signal for 1% supply variations. For this reason, these disturbance variables were eliminated from further consideration. The numerical values for the error coefficients, as calculated using the Stepwise Regression Analyses of six sets of test data, are given in Table 3.1. The Kg , K^ , Kg , K,, and KQ coefficients were uncovered in the test data by an analysis of the residuals. They are reasoned to be the results of the torques exerted on the float by the magnetic suspension forces when the float is not at its zero suspension-force point26.

TABLE 3.1

Parameter Variation Test Results

Term

ps Pi po vs

V|

v0

ws w, KB

KA

Ks K|

KO GSI GIS

Name

SAT

IAT

OAT

SATDOT

IATDOT

OATDOT

SARDOT

IARDOT

MIAR

MSAR

FSAT

FIAT

FOAT

TGSA

TGIA

Coefficient

0.0565

-0 .0498

0.0222

- 0 . 0 1 9 5

0.0333

-0 .00162

-0 .0645

0.0314

-0 .6127

2.886

- 0 . 1 8 8 9

0.0964

0.00626

- 1 1 . 7 4

11.52

Units

drums/// '

drums///'

drums///"

drums/u"/hr

drums/u"/hr

drums///'/hr

drums/sec/hr

drums/s'ec/hr

drums/sTc2

drums/sec2

drums///'

drums/// '

drums/// '

drums/°Fg

drums/°Fg

A plot of the gyro drift rate recorded during the tumble test is presented in Figure 3.4. Three orientations of the gyro with respect to Earth rate and gravity are shown. Three cycles of both clockwise and counterclockwise table rotations were analyzed. A least-squares solution for nine specific force sensitive drift coefficients and three misalignment angles of the gyro input axis into the table axis results in the values given in Column A of Table 3.2. The Standard Error of the Coefficients and the Student T test values are also given in this table.

34

3.8 t

0 . I 0 . 3 0 .5 0 . 7 0 . . 1.1 TIME (HOUHS)

1.3 l . S l . r * 1.*)

Fig.3.3 Parameter variation test, spin axis up

A comparison of the t-values with 2.58 indicates that five of the coefficients (indicated by an asterisk) are not significant at the 99% confidence level. A solution for DQQ cannot be obtained along with values for D,* and Dgg unless the magnitude of the specific force input vector is varied. A look at the residual errors for this model in Figure 3.5 shows that certain deterministic components remain. These are predominantly the first and second harmonics of table angle that cannot be accounted for by the specific force inputs alone. The inclusion of all third-order specific force terms in the regression equation had no noticeable effect on the solution.

Due to the high degree of correlation involved, it is not possible to analytically separate those components of the drift rate signal which are generated by float motion and thermal disturbances. It is possible, however, to use previously determined values of these coefficients to compensate the drift rate data before the least-squares fit is performed. The values obtained for the specific force coefficients using this procedure are given in Column B of Table 3.2.

35

8 . 0

XJI

cr

< tr

I O

>

o ••

- * • . o • •

- 8 . O ••

l . O 2 . 0 3 . O *».0 Tlf-IE (HOURS)

Fig.3.4 Tumble test drift rate

s.o b . O 7 . Q

TABLE 3.2

Comparison of Regression Analysis Results for Continuous Tumble Test

Dp

Di

Do DS

Dn

DOO

DSS

Dis D|0

DOS MAI

MA2

MA3

S.E.

R2

Units

drums

drums/g

drums/g

drums/g

drums/g2

drums/g2

drums/g2

drums/g2

drums/g2

drums/g2

drum, sec

drum.sec

drum.sec

drums

%

A (lassical Model

0.6899

-0 .0137*

-1 .5050

0.4308

- 0 . 9 4 4 4 *

- 0 . 8 1 9 3 *

-3 .3530

-0 .1298*

-3 .2110

- 0 . 8 *

-1832 .0

253.0

1.0235

95.94%

S.E.

0.2309

0.1059

0.0817

0.1014

0.3850

0.4325

0.2152

0.1597

0.1456

19.5

19.5

19.5

/- Value

2.99

0.17

18.42

4.25

2.45

1.89

15.57

0.81

22.09

0.04

93.73

12.94

B Extended

Model

- 0 . 2 4 3 3 *

0.2485

- 3 . 8 7 7 8

0.6577

0.5060

0.8535

-2 .4633

0.0352*

- 0 . 3 2 0 6

- 1 3 2 . 5

-1175 .0

527.0

0.4506

99.23%

S.E.

0.1011

0.0467

0.0359

0.0447

0.1702

0.1908

0.0947

0.0703

0.0642

8.6

8.6

8.6

t- Value

2.42

5.33

107.8

14.7

2.97

4.47

25.9

0.50

4.99

15.4

136.6

61.3

* Not significant at the 1% Confidence Level.

A comparison of Columns A and B shows that the test standard error was reduced from 1.0235 drums to 0.4506 drums by expanding the model to include the additional error terms. A total of 99.23% of the sum of squares of the drift rate signal was accounted for by this model. A comparison of the residuals plotted in Figures 3.5 and 3.6 graphically shows the improvement obtained in the data fit using the extended error model.

36

i . O

D Of 4 O

- 4 . 0

- « . O

l . O 2 . 0 J . O •». O

T i r - e ( H O U R S I

S . O 6 . 0 ' . a

Fig.3.5 Residuals from tumble test using classical error model

The most noteworthy differences in the magnitudes of the drift coefficients occurred in the values of D o s

and D,s . Both of these "compliance" coefficients were found to be smaller than those obtained from the classical error model solution. This is more in agreement with the results obtained from linear vibration tests which generally show smaller magnitudes for the compliance coefficients than obtained from laboratory testing. Values obtained for the other compliance coefficients (D*j,Dss, and DQQ) **re highly unreliable as indicated by the small t-values obtained as a measure of their significance.

».o -

J : S

1.0

o . -v \^ / v wv «*• vv*vvANW lv\^^

-*».o

-» . o

1.0 fl. o J . o *».o TiT-E (HOUMS)

S.O t . o 7 . 0

Fig.3.6 Residuals from tumble test using extended error model

The magnitudes of the "unbalance" coefficients ( D , , D Q , and Dg) were affected by the presence of thermal gradients and changes in float position. The sizeable change that occurred in the value of DQ is not unusual for

37

this coefficient, since its value is highly dependent on the orientations used for its solution. Additional analysis indicates that the translation of the float along the output axis has a dominant effect on the calculated value for DQ (Ref.26).

The fact that the extended error model provides a more accurate description of the gyro drift rate behavior during a tumble test is evidenced by the removal of systematic quantities from the residuals. Although the final residuals are not completely random, the Probability Density Function shown in Figure 3.7 indicates that they are approaching a normal distribution.

The relative importance of any one gyro error model coefficient depends directly on the magnitude of the forcing variable experienced by the gyro. These values can be estimated for a particular mission and an error budget constructed to compare the contributions of each error source. Instead of doing this for some arbitrary mission environment, a table of relative importance has been prepared for errors observed during laboratory testing of this particular test specimen.

35.c • — 3-..0 . i n 33.0 . Ill 32.0 . Ill 31.0 . 111 30.0 • 111 111 29.0 . Ill 111 28.0 , —-111 111 27.0 . llllll 111 26.0 . llllll 111 25.0 • llllll 111 2-V.O . I l l l l l l l l l l l 111 ?3.o . 111111 m i n i m u m — 22.0 . l i n n m i n i m i n m n 21.0 . m m 111111111111111111 20.0 *> m m — m i n i m i n n 111 19.0 . n 11 11 11111 l 1 1 1 1 1 1 1 n i i n n iB.o . — i i u u u m n 1111 ii 111111 n 17.0 . I m 111 11 1 I l l l 1 1 I 11 1 1 II 1 1 1 1 I 11 — i6 .o . i 11 m i n i n n m i n i n i i n i u 111 i5.n • — — m u i i i i u i u m u u i u i i u u m i«v.o . m m i m i n n m m i m i i n 11 m n m 13.0 . u i m — i m m n m i m m m i m i l n 111 12.0 . i l l i i m i m m i n m m m m i i m i i i n n — u . o . — i n — 11 m m i i i n n i i n n 11 n m 1111 n u u i — i l l — — 10.0 4- i m i n i i n m m i m i n n i n i n i n i m n i n i i n i n n u n — i n <>.o . i m n n m n n i n m i m i n i m m m i 111 i n i n n i n i n i i i i i i i s.n . m i n m m n n m n m i n n n i n i i i n n n m m n m i n i u m 7.0 . — m i n i n n m n n i i n n i n n i n n n n n i n n n i n i — m i n i u m 6.0 . — i l l m n n n n n m m n n i i n m i m m i i n i n n n n i i n m i u r n m 5.0 * m i n i n i m m m i n i n m u m m u u n u u m i n n i u i i u i i i u i n m — •v.n . m — i n n n i n i i n m m i 11 n m i n n i n 1111 m 111 i n m m m n i i n n n u n 3.0 . m i n i i i - - - i n i n i i n m m i i n m i m m i i n i n n i n i i n i i n i n n u m n i m i n 2.0 . — u u i u u n i i n i m m m n 111 i n m i n i n i i n i n 11111 n n 1111 n m i u i m i m i n — i n i .o . m m u i i u u i u i u m m u i n n i m i i n m m n m m m m m m i m i m m m m m i n

-1 .0 -0 .8 -0 .7 -0 .6 -0.5 - " . » -0.2 - ( , , ! J.(j 0.2 I . ) 0.4 1.5 0.7 0.H " . <* 1.0

Fig.3.7 Probability density function test residuals using extended error model

Table 3.3 shows the error coefficient, its magnitude, the maximum value for the forcing function in a laboratory test environment and the total error term contribution in drums. It can be seen from this table that several of the error model coefficients added to the error model for this gyro are relatively more important than some of the classical specific force sensitive coefficients.

3.2. 7 Summary and Conclusions

The results of an experimental investigation into the nature of random gyro drift rate disturbances have been reported in this paper. Physical models were proposed to describe the behavior of the non-ideal gyro output signal under the influence of temperature variations, float disturbances, and input power variations. A methodology was developed for identifying the importance of these proposed error sources and accounting for their undesirable effects by output signal compensation. New gyro test procedures were employed and test data analyzed in order to extract the numerical values for the coefficients of the proposed mathematical error model.

The data that were analyzed indicate that a large portion of the uncertainty which is observed in the output signal of a gyro stems from deterministic error sources. Consequently, the accuracy of a gyro can be considerably improved by investigating the fundamental sources of input disturbances and by understanding of the physical mechanisms by which these disturbances manifest themselves as torques about the gyro output axis. The approach taken here was to empirically confirm or disprove the presence of proposed error model terms and to closely examine the remaining uncertainty for additional relationships.

38

TABLE 3.3

Comparison of the Relative Magnitudes of the Gyro Error Model Terms in a Laboratory Test Environment

Relative Priority

1 -i

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18 - — - .

Coefficient

D 0

Dis C'S1 GIS ps V|

DS po DSS

vs

DM

Dl DF

v0

DOS

ws pl

w.

Sensitivity

- 3 . 8 8

- 2 . 4 6

- 1 1 . 7 4

11.52

- 0 . 1 3 3

0.033

0.658

0.026

0.854

- 0 . 0 1 9

0.506

0.248

- 0 . 2 4 3

- 0 . 0 0 1 6

-0 .321

- 0 . 0 6 4

- 0 . 0 5 0

0.0314

Units

drums/g

drums/g2

drums/°Fg

drums/°Fg

drums/// '

drums/u"/hr

drums/g

drums/// '

drums/g2

drums//* "/hr

drums/g2

drums/g

drums

drums/u/hr

drums/g2

drums/scc/hi

drums/// '

drums/s'ec/hr

Forcing Function Magnitude

±1.0g

±0.5 g2

±0.1°Fg

±0.1°Fg

±5 .0 / / '

±20.0u"/hr

± l.Og

±20.0 u"

±0.5 g2

±20.0// ' /hr

±5.0g2

±1.0g

±1.0

±100.0// ' /hr

±0.5 g2

± 1.0 sTc/hr

±0.5u"

± 1.0 sec/hr

Drift Rate Error (Drums)

±3.88

±1.23

±1.17

±1.15

±0.67

±0.66

±0.65

±0.52

±0.43

±0.38

±0.25

±0.25

±0.24

±0.16

±0.16

±0.06

±0.03

±0.03

The most important conclusion to be derived from this investigation is that an extension of the gyro error model to include additional error terms is feasible and indeed desirable. The use of a more complete error model facilitates a more accurate determination of the gyro drift coefficients and can be used for compensation, thus increasing the accuracy of a particular instrument. This pilot investigation has confirmed some existing theories on gyro torque-generating mechanisms, postulated and evaluated new ones, and established a methodology for continued research to empirically derive the appropriate relationships between the gyro output signal and interfering input disturbances.

i . i Precision Centrifuge Test

3.3.1 Introduction

Improving the accuracy of inertial instruments for missile applications requires an increased understanding of the instrument's performance over time intervals which are quite short with respect to typical boost flight times. Additionally, a better knowledge of the behavior of the instrument's specific force sensitive coefficients at higher acceleration levels is essential. In this Section a centrifuge test is proposed that permits a solution for all of the classical gyro error model coefficients every 8 to 10 seconds. In this way the short-term variations in the gyro drift coefficients can be observed. This is a considerable improvement over other methods that normally take several hours to obtain independent solutions. In addition, the centrifuge can provide input specific force levels that are appreciably greater than the one-g input provided by normal laboratory tests. This not only gives a greater gyro output signal due to the specific force sensitive terms, but also exercises the instrument error terms in a specific force region more representative of the boost flight profile. Thus, an improved confidence level in the magnitude of the error coefficients can be achieved along with a better measurement of coefficient linearity. Some of the limitations and special considerations that must be given to the implementation of the proposed test are also given.

3.3.2 Test Procedures

The basic procedure proposed to measure the gyro error model coefficients on a precision centrifuge is to continuously reorient the gyro axes with respect to the centrifuge acceleration and gravity. This test requires the use of a precise counter-rotating platform (CRP) in order that the centrifuge angular rate is eliminated as an input to the sensitive gyro axis. This arrangement provides a low frequency, sinusoidally varying acceleration profile to

39

the gyro under test. By rotating the instrument about one of its horizontal axes at some multiple of the centrifuge angular rate, continuously varying acceleration inputs can be supplied to all three gyro axes.

Figure 3.8 shows the orientation of the gyro with respect to the centrifuge axes at the start of the data sampling process. For this situation:

0 = Wct

where 0 is the centrifuge rotation angle and Wc is the centrifuge angular rate.

(3.24)

COUNTER-ROTATING PLATFORM

C

Fig.3.8 Gyro orientation for centrifuge test

For synchronous operation of the counter-rotating platforms (CRP)

0 = 0 ' . (3.25)

By restricting the rotation of the gyro about its output axis (7) to be a multiple of the centrifuge rotation angle (0), the following relationship holds.

7 = n0 and 0 = m0 . (3.26)

The starting position occurs when 0 = 0 ° for the gyro input axis north, and 7 = 0 ° for the gyro spin axis up and input axis horizontal. Gyro data is taken at fixed intervals of centrifuge rotation angle after appropriate analog and digital filtering have been performed.

3.3.3 The Gyro Performance Model

A description of the gyro's output signal, Wd , can be written in terms of its input angular rate, to, , and a linear expansion of its specific force inputs, f, along its three principal axes. A widely accepted form for this performance model is given below.

Wd = co, + Dp + D,f, + D0f0 + Dsfs + D„ff + Doo-0 + DSS*S + D,o¥o + D i s ¥ s + D0S*bfS + oco . (3.27)

Specific force, as defined in Reference 27, is the vector sum of inertial acceleration forces and the gravitational field forces.

f = a - g . (3.28)

40

For the starting position shown in Figure 3.8, the components of Equation (3.28) expressed in the gyro reference frame are

fo =

ac

0

0 -

0

0

—g

= ac

0

g

where the coordinate axes are along the gyro input, output and spin axes, respectively.

For rotations of the centrifuge about 0 , the coordinate transformation is written

R* =

cos 0 sin 0 0 —sin 0 cos 0 0

0 0 1

For rotations of the rate table about y , the coordinate transformation is written

cos 7 0 sin 7 0 1 0

—sin 7 0 cos y

The combined effect of these rotations is

Letting y = n<p and 0 = m<b we have

M = R ^ O

r i r 1

f. -fl fo fs

ac cos 7 cos 0 + g sin 7 —ac sin 0

—ac sin 7 cos 0 + g cos y

, we have

f. " fl fo fs

=

ac cos n<t> cos mxp + g sin n0 —ac sin m4>

—ac sin n<j> cos m<j> + g cos n0

The components of angular rate about the gyro reference axes at the starting position are

(Aim

wE

WE

cos

wp

sin

L

L

(3.29)

(3.30)

(3.31)

(3.32)

(3.33)

(3.34)

(3.35)

where W£ is the magnitude of earth rate, Wp is the rate table angular rate, and L is the local latitude angle.

For rotations about y (gyro output axis)

BU

cos 7 0 sin 7

0 1 0

— sin 7 0 cos 7

(3.36)

For rotations about o. , which represents a misalignment angle about the gyro spin axis

* a —

cos a sin a 0 —sin a cosa 0

0 0 1

(3.37)

The combined effect of 7 and a is

co, = RQI^CO,- (3.38)

41

t o 0

Wp cos L cos 7 + Wp sin L sin 7 — Wptv

a(WE cos L cos 7 + Wp sin L sin 7)Wp

—Wp cos L sin 7 + Wp sin L cos 7

(3.39)

where cos a = 1 and sin a = a has been substituted for small values of a .

Angular rates about the gyro input axis must be calculated and used to compensate the gyro output signal if the effects of the error terms are to be observed. This quantity is given in Equation (3.40).

to, = Wp cos (7 — L) — Wptv

to, = Wp cos (n<j> — L) — W pa.

Using the following trigonometric substitutions

sin m<j> cos rnp = \ [sin (m + n)<p + sin (m — n ) 0 |

cos m0 sin ntj> = \ [sin (m + n)ip — sin (m — n)$)

cos mxp cos mp = \ (cos (m + n)<p + cos (m — n)<p]

sin mxp sin n<j> = \ [cos (m — n)<j> — cos (m + n)<j>]

the coordinate functions for the performance model given in Equation (3.27) can be written as follows:

a, f, = — [cos (m + n)(j> + cos (m — n)tf>) + g sin n<j>

fG = — ac sin m0

fs = — — [sin (m + n)<p] + g cos n<j>

3 3 3 3 f? = — + — cos 2n<p + — cos 2m<p + — [cos 2(m + n)0 + cos 2(m — n)ij>] 4-1 4 4 4 8

+ -%- [sin (m + 2n)# — sin (m — 2n)<p] + — — — cos 2n0

(3.40)

(3.41)

•6

• 1

fifo

fofs

— — — cos 2m<t> 2 2

a a a a _£ - _c Cos 2n<p + — cos 2m<t> - — [cos (2n + 2m)«i> + cos (2m - 2n)0] -4 4 4 8

ga B 2 22

— —- [sin (m + 2n)0 — sin (m — 2n)$] + — + — cos 2n0

— [sin (2m + n)0 + sin (2m — n)0] — — [cos (m — n)0 — cos (m + n)<j>] 4 2

3 il *'

— (cos (2m — n)0 — cos (2m + n)0) — -^- [sin (m + n)<j> + sin (m — n)4>]

f,fs = - -^ sin 2n0 - - | [sin (2m + 2n)0 + sin (2m - 2n)0] +

+ — [cos (2m + 2n)0 + cos (m - 2n)0]

g2 sin 2n0

(3.42)

42

Substituting the above relationships into the gyro performance model given in Equation (3.27) and separating the terms by harmonics gives the following expression:

Wd = - W p a + DF + D„ ( j + 3 ) + DSS ( f + f ) + DQO f + (DSS " Dn) ^ sin (m - 2n)0 +

+ (D,g + WEsin L) sin n0 + ( D O S ^ + Ds y j sin (m - n)cp - DIS f - ^ - j sin 2n0 -

( ^ + DS y ) sin (m + n)0 -Doac sin mtp — D|§ — sin (2m — 2n)ij> — DQS 8

gac — D,0 — sin (2m — n)0 — (D s s — D„) ~2z sin (m + 2n)0 — D*Q — sin (2m + n)0

— D,s — sin (2m + 2n)0 -I- D,s —- cos (m — 2n)0 + (Dsg + Wp cos L) cos n0 + 8 2

+ fD, - DI0 ! £j cos (m - n)0 + (D„ - Dss) ( ^ - Lj cos 2n0 +

+ (D„ - Dss) cos (2m - 2n)0 + (D, + Dl0 j cos (m + n)0 +

• cos (m + 2n)0 + HD,, J - D 0 0 j + 2

f) - J j cos 2m0 - D o s J cos (2m + n)0 + (D„ - D s s) ^ cos (2m + 2n)0

> (3-43)

There are many ratios of m to n (such as 5:1, 5:2, 6:5, 7:2, 7:5, and 10:1) that will minimize the correlation between the various error coefficients. For the purposes of this analysis a ratio of 5:1 is chosen. This requires a total of five revolutions of the centrifuge arm for a complete solution while the rate table axis rotates once.

3.3.4 A Solution for the Drift Coefficients

A solution for the drift coefficients can be obtained from Equation (3.43) by separating the various harmonics using a Fourier Analysis of the gyro output data. The magnitude of these harmonics can then be algebraically combined to obtain the respective drift coefficient values. In order to obtain a solution for all coefficients it is necessary to assume that the value for DQO *S zero. This assumption (or the assumption that either Dgg or D,, is zero) is normally used when reducing data obtained from one-g tumble tests, including the two-axis tumble test described in Reference 20.

From Equation (3.43) the Fourier coefficients can be separated and represented as follows:

A0 = - W p a + Dp + DSS (% + - Q

A, = D,g + WE sin L

+ D00 y + D„ M

A3 = D S S «J- Di 8ac > (3.45) cont'd

A4 "

A5 =

A6 =

Dos f + DS f

~D 0 a c

- D 0 S ? - D S ^

43

A-, =

A8 =

A, =

A =

A n =

A, 2 =

DSs - y + DII - . j -

- D a '

-D,of

0

D 3 ' Dio 7

D "2

B, - Dsg + WE cos L

..=-DSs(H)+.,eH) B,

B„

B5

=

=

=

Dis

D,

0

gac

a c _ 2 Dio

gac

2

B. = D, & + D,o ^

B, = DIS g*ic

aj 8

B. : - D S S "7 + D» ?

B, = D 0 S

B,o = -Doo f + Dss f + D„ -J

B n - _ D 0 S j

B„ = - °SS T + D „ ac ^ ^ 2c

(3.45) cont'd

J

A solution for the nine drift coefficients is obtained from Equation (3.45) as shown below, where g = 1 has been substituted.

DF = A0 + Wpa + 2(Bg + B12) (• + ^ ) + B10

D . = B* + B* = A, - WE sin L

n - - A '

> (3.46) cont'd

44

A — A D s = — = B, - WE cos L

ac

DII " - i (fi8 + B,o + B12) •#1

DSS =

B.o -

D i e =

- 2 ac

- 4 A q

ac

- 4 A 2

B,o + B1 2)

- 4 A . . B, — BJ,

ac ac

- S A g - 8 A „

- ( A 4 + A6) 4B9 - 4 B „ Dos : -2 Ti—

2B3 2B,

(3.46) cont'd

From Equations (3.46) it can be seen that redundant solutions exist for some of the drift coefficients. A preferred method for obtaining the "best estimate" of the coefficients in this case would be to calculate the least-squares fit of the performance model to the test data. This yields the maximum likelihood, minimum variance estimate if the performance model is complete. Since there is a good possibility that this may not be the case for testing a gyro in a high acceleration and a high angular rate environment, the Fourier analysis method may provide better insight into the determination of significant error sources omitted from the model. With this thought in mind several observations can be made from Equations (3.46).

(i) The estimated Dp coefficient will be highly unreliable. This is not a severe limitation to the test since its primary purpose is to evaluate the short term instabilities in the specific force sensitive drift coefficients. Instabilities in the fixed drift term can best be evaluated in a static, one-g environment.

(ii) A solution for all drift coefficients can be obtained without making any earth rate compensation to the test data. This is possible because the D* and Ds coefficients can be calculated in two different ways.

(iii) All of the specific force squared sensitive coefficients can be calculated from Fourier Coefficients having magnitudes which are proportional to centrifuge acceleration squared. This permits a greater confidence level in the coefficient estimates since the forcing function has a larger magnitude.

Therefore, the desired equations to be mechanized for the calculation of eight gyro drift coefficients every five rotations of the main centrifuge arm are:

Di =

D0 =

DS =

Dll =

B4 + B6

ac

- A 5

ac

A 4 - A 6

(B8 + B I 0 4- B12)

DQO = 0 (assumption)

- 2 ( B , 4- B12 - B,0) Dss -i

ac

Dis

DlO =

- 4 ( A , + A12)

- 2 ( A , + A,,)

2(B9 4-B..) DOS 7i

(3.47)

45

A solution for these coefficients will be obtained every 8.15 seconds as calculated below.

360 degrees/cycle 5 cycles 221 degrees/second solution

8.15 seconds/solution

In order to calculate the magnitudes of the Fourier coefficients up to the twelfth harmonic and to obtain sufficient data density, a sampling interval of 5 degrees of centrifuge rotation should be used. This allows for 72 samples per revolution and 360 samples per solution. At this interval the data sampling rate is calculated to be:

221 degrees/second = 44 records/second

5 degrees/record

where a data record will consist of several samples, one for each variable of interest. If twenty samples are taken for each record, a throughput rate of 880 samples per second is required of the data acquisition system. A certain amount of on-line filtering and pre-processing is required before storage, if data storage device saturation is to be avoided for long testing intervals.

3.3.5 Test Error Analysis

An attempt to perform the above described test will result in several major error sources unless careful attention is given to accurate alignment, rate control and positioning. An initial look at some of these error sources is given in this section in order to establish test procedure requirements for achieving the desired test accuracy.

A convenient centrifuge input acceleration of lOg's is selected for the purpose of this analysis. Using a 260" radius arm, the centrifuge angular rate is:

Wc = 1127 j - ^ = 221°/sec (3.48)

where R is the radius in inches, ac the centrifuge acceleration in g's, and Wc the angular rate in degrees/second.

The rate table angular rate is then calculated to be:

n 771 W. = - Wc = — =. 44°/sec (3.49)

v m 5

A misalignment of the gyro input axis into the rate table rotation vector will cause a gyro bias signal to be read. This should be held within limits in order to prevent a large output axis rotation angle and also to prevent saturating the desired error signal. Constraining this bias rate to be less than 10°/hr requires an alignment accuracy of

Bias(°/hr) (55.5) (10) a(sec) = 55 .5—-3-—- = — = 12.5 sec. (3.50)

Wp ( /sec) 44

More critical to the accuracy of the coefficient determination themselves is the drive accuracy of the rate table. Higher frequency variations in the table rate will be filtered out, but variations in the table rate which are at multiples of the centrifuge rotational frequency will cause errors in the calculated drift coefficient values. For a gyro input axis misalignment angle of 12.5 sec, the rate table drive accuracy (at the harmonic frequencies) required to induce less than 1% error in the coefficients is given below:

5Wp(sina) - (0.01 )D(°/hr/g2) (100 g2)

5Wp(°/hr) = (1.6 x I04)D (3.51)

where D is the magnitude of the drift coefficient to be determined and 6Wp is the zero-to-peak variations in the magnitude of the table rate.

If we assume that D is on the order of 0.05°/hr/g2, then 6Wp = 800°/hr (0.22°/sec). For a total rate of 44°/sec this amounts to an allowable rate table variation of 0.22/44, or about 0.5%.

An important factor to be considered in the conduction of this test is the frequency response of the gyro torque rebalance loop. For the rates and ratio assumed in this analysis, the highest harmonic of interest (twelfth) will occur at 2.4 Hz. The amplitude ratio and phase shift of a first-order torque rebalance loop with a break frequency co0 can be calculated from Equations (3.52) and (3.53).

AR(to) = . "» , (3.52) V col + w

4<J

0(to) = tan"1 — . (3.53) " o

If we desire to constrain the phase shift in the recorded data at 2.4 Hz to one degree then

to (2vr)(2.4) to0 >

tan 0 tan lc

Wo > —7T--.— = 860 rad/sec 0 0.01746

860 fn > = 137 Hz .

2vr ' 0

(3.54)

Since this frequency response is considerably higher than most rebalance loops are able to provide, it will be necessary to accurately measure the amplitude ratio and phase shift outputs at each of the harmonic frequencies of interest. The test data can then be appropriately compensated to account for these errors.

Another possible source of error that may be observed is the apparent input axis misalignment effect caused by angular rates applied to the gyro output axis. A description of this effect can be found in Reference 28. This effect should not restrict the usefulness of the proposed test in any way since it will appear as a constant bias effect. If the magnitude of this bias is unacceptable or if a two degree-of-freedom gyro is being tested, then a variation of the same test may be performed by rotating the gyro about its spin axis. The drift coefficient solution will remain the same except that the O and S subscripts will be interchanged.

A deviation of the counter-rotating platform from its true synchronous rate will cause an angular rate error to be sensed by the gyro input axis. It will also generate a misalignment of the gyro axes from their calculated position. For this reason it will be essential to measure the angular positions of the centrifuge arm, CRP and rate table in a continuous manner in order that their angular rate and position can be used to compensate for these known error effects.

Additional error sources not considered here are certain to be evidenced during testing. Adequate instrumentation should be provided during testing in order that the total environment can be examined for cause and effect relation­ships. This can then lead to performance model modification or a change in the test procedure in order to achieve meaningful test results. As a minimum the following test instrumentation is recommended for any pilot studies that might be conducted using these test procedures.

(i) Three linear accelerometers mounted coaxially with the principal gyro axes in order to identify any input specific forces to the gyro which have not to be accounted for in the analytical model.

(ii) Temperature gradient sensors along the gyro input, spin, and output axes.

(iii) Gyro float motion monitors along the input, spin and output axes if such readout is available.

(iv) Angular position of the centrifuge arm, CRP and rate table.

(v) Time.

(vi) Total gyro wheel power.

In addition to the required sensors the following test instrumentation is essential to the performance of these tests.

(i) A high-speed data acquisition system capable of multiplexing and analog-to-digital converting of 20 or more signals at the rate of 1000 samples/second or greater.

(ii) Computer processor to perform on-line filtering and pre-processing of data before storage and/or display.

(iii) Mass storage device such as a 7-track magnetic tape recorder or moving head disk.

(iv) High-speed line printer and plotter (or graphical display device).

3.3.6 Summary

A novel approach has been suggested for the evaluation of a gyroscope on a precision centrifuge using a counter-rotating platform. An analysis of this procedure indicates that eight of the conventional specific force and specific force squared sensitive drift coefficients can be separated after each five passes of the centrifuge arm. This amounts to independently derived solutions every 8 seconds and permits a measurement of the short-term character­istics of these drift coefficients. The addition of recursive estimation techniques will yield smoothed solutions at each sampling instant (about 44 times a second).

47

An advantage of the described test over one-g laboratory tests for coefficient determination is that higher acceleration input levels are used. This provides for a larger measurement signal and increased confidence in the estimated coefficient values. It can also be used to measure coefficient linearity by running the same test at different acceleration levels.

Disadvantages of this test center around incomplete information about the nature of the inputs provided by the centrifuge test equipment and a lack of experience in evaluating a gyroscope in a high angular rate and acceleration environment. For these reasons it is essential that the gyro and the test environment be extensively instrumented initially, so that error sources can be identified and modifications made to the analytical models.

3.4 Vibration Testing

3.4.1 Introduction

3.4.1.1 Background

The intended use of precision inertial gyros in a high-g environment has resulted in considerable attention being focused on the linear acceleration sensitive terms of the gyro error model in a greater than one-g test environment. The two primary test devices which have been used to provide this input are the precision centrifuge with a counter-rotating platform and the precision linear vibrator. The precision centrifuge can apply a sustained linear acceleration over relatively long time periods, but it also generates a relatively large angular velocity to which the gyro is extremely sensitive. The linear vibrator can provide a sinusoidal linear acceleration over a fairly wide bandwidth (5 to 2000 Hz) and a large amplitude range (up to lOOg's). Even the most precise linear vibration device is not without problems however. These are primarily caused by structural resonances, large cross-axis vibration levels, and angular vibration inputs and offsets. In spite of these limitations, some very valuable information has been obtained from linear vibration testing concerning the nature of drift coefficients that are functions of accelerations squared and of the higher even-powers of acceleration.

Recognizing the important need for ascertaining the performance of a precision gyro in a high-g, vibratory environment and the inherent limitations of trying to extrapolate one-g laboratory test results to predict this performance, some reliable high-g test methods must be developed to meet these requirements. The long-standing disagreements which have been observed in the drift coefficients that are obtained from tumble, servo and linear vibration testing must be resolved before any accurate estimates of error compensation values can be made. In this section I shall postulate some reasons for these discrepancies and shall offer some suggestions for making improve­ments in the methods used to obtain linear vibration test data on gyroscopes. After several years of making refine­ments in the design and construction of linear vibration test equipment to achieve the precision and accuracy required to carry out well-established test methods, we have reached a point where further improvements are extremely costly and time-consuming. It appears that linear vibration testing is in dire need of some novel ideas and some creative test design and analysis procedures.

3.4.1.2 Limitations

A description of modern linear vibration test equipment and procedures is contained in Reference 2. These methods are designed to investigate only the second and higher, even-powers of acceleration sensitive gyro drift. The data from these tests are obtained from the time-averaged or d.c. value of the gyro's output signal. Consequently, the greater portion of the test data, namely the a.c. component, is discarded. This is not only extremely wasteful of valuable test time, but also presents the test engineer with a most difficult task of trying to extract the desired coefficient values from the d.c. component, which is the one most susceptible to drift, bias errors, mechanical distortions and other yet undefined gyro error phenomena.

The most successful test designs which have been developed attempt to separate the desired error terms in frequency and/or phase from the undesirable interfering inputs. Recent improvements which have been made in the frequency response of gyro servo amplifiers and the availability of high speed data acquisition systems makes such an approach applicable to linear vibration testing. Improved linear and angular motion sensing devices are available to precisely monitor the test environment instead of using calculated values based on the assumption that the applied input is consistent and correct. Inexpensive analog multiplexers and mini-computers are available to acquire, preprocess and store this multi-channel data recording and analysis operation. This basic approach of recording and processing all of the gyro's output signal during linear vibration would allow the investigation of the acceleration-sensitive drift which is the result of gyro mass unbalance. In addition, all of the higher, odd-powers of acceleration sensitive drift can be obtained.

Most of the precision linear vibration test data which have been obtained to date have been concentrated at a few selected frequencies. This is mostly due to the fact that it takes so long to (i) align the test specimen, (ii) average the output signal for several minutes, (iii) take both static and dynamic readings, and (iv) reposition the instrument to several test orientations to separate the six compliance coefficients. If a test can be designed to obtain the desired information from just several cycles of the input vibration frequencies then perhaps some or all of the following possibilities can be incorporated; (i) Rapidly sweep through many different frequencies by employing the use of

48

random input excitation, (ii) continuously reorient the gyro during vibration in order to provide vibratory accelera­tions to all gyro axes simultaneously, and (iii) process the data continuously in real time as the test is performed. Assuming that a test can be designed that would separate and yield continuous estimates of all of the desired drift coefficients, then the test could be continued indefinitely to give (i) a measure of the short-term time variation of the estimated values or (ii) a running average of the coefficients in order to improve accuracy and diminish the effects of measurement uncertainties.

A change in the concept of how vibration tests are conducted opens many new possibilities for considerable improvements. Some new concepts have already been explored for angular vibration testing of strap-down gyros in Reference 30. In this analytical study the higher frequency harmonics are used to extract the fundamental design parameters which form the basis for the angular vibration sensitive coefficients of a gyro. Several experi­mental schemes were proposed and a set of feasibility tests were recommended to identify the most significant sources of test error.

3.4.1.3 Recommendations

Several improvement areas for successful linear vibration testing warrant further investigation and experimentation. Some of the limitations in present test equipment can be overcome by careful monitoring, test structuring, and improved engineering. The usual testing fixture consists of an electrodynamic vibrator and slip table combination mounted on a large seismic mass which is supported by four air springs. The purpose of the seismic mass is to absorb the energy which is being expended by the shaker, while the air springs provide high test-system inertia and high-frequency isolation from base motion disturbances. The combination of some mass unbalance normal to the axis of vibration and the soft support provided by the air springs results in considerable rocking motion on the test platform at low frequencies. Since a gyro is designed to be most sensitive to angular motion, this interfering input can provide a major contribution to the gyro's output signal. One solution which should sizeably reduce the magnitude of this problem is to use two opposing shakers. If equal action and reaction forces are generated, then the seismic mass does not have to absorb any of the applied force. This would provide a more uniform linear motion to the test specimen minimizing the rocking effects, reduce the requirement for precise package balancing about the vibration axis, and could even permit simultaneous testing of two instruments. The opposing motion can be achieved by the use of the same excitation signal to both shakers using matched control accelerometers.

Another possible improvement would be the use of an on-line digital computer to generate the control signals for the shaker. In this way the servo loop compensation parameters can be easily varied to accommodate shaker and slip table variations and changes in package loading. A computer would also be capable of generating almost any random input spectrum that could be described mathematically. Thus, some very expensive analog shaping networks and noise generators are eliminated in the process. Almost unlimited versatility is available with the computer for automatic test sequencing of amplitude, frequency, filtering parameters, and data acquisition. Either a digital magnetic tape or a disc storage system can be used to store many data channels at a relatively high sampling rate. A high-speed data acquisition system will provide the necessary multiplexing, analog-to-digital conversion, and input to the computer. Depending on how extensively the computer is involved in controlling the shakers, it could be used for real-time processing and display of quick-look gyro test data. Thus, the effect of changes to the test setup could be observed immediately, instead of waiting for the test data to be reduced many hours later on a large general purpose machine.

The development of the Fast Fourier Transform and its implementation in mini-computers offers some extremely powerful techniques for data reduction and analysis. Random vibration inputs can be used to excite the test specimen over a wide frequency range. The calculation of cross-spectral densities between the input and gyro output signal provides a calculation of the gyro's drift coefficients as a function of frequency over this same range. This wealth of new information should yield new insights into some of the problems which have plagued linear vibration testing in the past.

Careful instrumentation and monitoring of the total motion of the test package while it is undergoing vibration excitation will eliminate the requirement for precise control of the test motion environment. The use of a compre­hensive dynamic gyro error model along with a measurement of the input motion will allow for compensation and simultaneous coefficient estimation. An array of high-quality linear accelerometers can be mounted on the gyro holding fixture. The six degrees-of-freedom of test fixture motion can be determined by algebratic manipulation of these measurements. Alternately, angular motion sensors can be employed to make direct rotational measurements. The future of gyro linear vibration testing however, probably lies in the use of a complete guidance system package. The outputs from the three gyros and the three accelerometers provide an excellent description of the test input motion. The error model coefficients for all six sensors can be determined at the same time. The models are then used to improve the original input motion measurements in a bootstrap technique. Judicious selection of test orientations should make the evaluation of all coefficients in all six error models possible. A very powerful test approach would be to use a continuously rotating test package orientation while undergoing a random linear and angular vibration environment. Cross power spectrums would again be used to extract the sensor error coefficient transfer functions from the multi-channel test data. Section 3.4.3 describes an analytical method that can be used to reduce this type of test data for a single instrument. It provides only a limited example of what can be done with today's instrumentation and computing equipment.

49

3.4.2 Linear Vibration Versus Tumble Test Compliance Coefficients

3.4.2.1 Compliance coefficient measurement

The disagreement that has been observed between the gyro compliance drift coefficients which are determined from linear vibration testing and from tumble testing can possibly be explained by the fact that an incomplete gyro error model is generally used in these solutions. The gyro error effects which have been briefly described in Section 3.2 can be of considerable importance during a one-g tumble test. In this type of test the level of the forcing function is very small since it cannot exceed 0.5 g-squared. The float motion errors, temperature effects and input power variations can however add a significant portion to the total drift rate output that is developed by the compliance coefficients in a 0.5 g-squared test environment. If these other disturbance parameters generate error terms that contain components which correlated with the drift generated by the compliance coefficients, then they would be erroneously assigned to the solution for the compliance coefficients.

As an example of how this might occur, a brief error analysis is presented in this Section. The gyro error effects that are associated with the motion of the float during tumble testing are used to indicate the nature and level of the errors that might occur in the compliance coefficient solutions. Many other error effects could be similarly considered.

3.4.2.2 Derivation of magnetic suspension torque coefficients

During tumble testing some off-floatation temperature will always exist at some point within the gyro. This is because it is extremely difficult to exactly float the gyro at both ends and along all axes simultaneously. As a result, there will be some small motion of the float with respect to the gyro case whenever the gyro is placed in a new orientation with respect to gravity. In the case of a magnetically-supported gyro the amount of this motion will be proportional to the stiffness of the magnetic-suspension circuit.

F = KD , (3.55)

where F is the (6x1) vector of generalized forces actingon the float center of buoyancy due to the unfloated mass; K is a (6 x 6) matrix of suspension stiffnesses; and D is the (6x1) vector of the generalized displacements of the float from a point of zero centering force.

For the case of a continuous tumble test, the float will be in a state of continuous motion. The error torques which result from a velocity of the float relative to the case can be sizeable if the float is not precisely centered in a perfectly circular case. These torques have been thoroughly described in Reference 21. For the case of a fixed position tumble test, the effects of float velocity will be reduced depending on the amount of settling time that is allowed before data sampling. The effects of a steady state position offset will however be evidenced in the test data.

In order to postulate how error torques about the gyro's output axis can result from its displacement from some reference point, a description of the float and case construction is given. There are essentially four points on the end of the gyro float which are of concern. These are (i) the float center of mass, (ii) the center of magnetic suspension forces, (iii) the center of buoyancy, and (iv) the geometric center. These points are physically tied together and move in unity as depicted in Figure 3.9. Besides a reference point (R) which is fixed with respect to the case, a point at which no magnetic suspension forces act on the float when point S coincides with it, is also indicated. The displacement of S from N is represented by the vector D in the figure.

In order that the origin of magnetic suspension torques about the gyro's output axis can be more easily identified, this discussion will be limited to two dimensions at one end of the float only. Whenever the float is not at its exact floatation temperature a buoyancy force (FR) is generated which acts through the float center of buoyancy along the direction of the local gravity vector (g). This force causes the float to be displaced from its point of zero suspension restoring force (N). The displacement continues until a magnetic suspension force (Fs) is developed which is equal to and opposite in direction to the buoyancy force. This force acts through the float center of magnetic suspension (S) along the local gravity vector, so that the summation of forces along that axis is equal to zero. However, since the center of magnetic suspension (S) is always displaced from the center of buoyancy (B) by an amount G a finite torque about the output axis will also result. This torque, of course, is counteracted by the gyro servo amplifier. The torque provided by the servo amplifier is represented as gyro drift rate.

In this same context the gyro mass unbalance coefficients^are the result of a displacement of the float center of mass (M) from the center of buoyancy (B) by an amount U . Magnetic suspension coefficients which are a result of the displacement G should be considered in the same way. Just as the mass unbalance coefficients are calibrated, so also can the magnetic suspension coefficients be calibrated. Through a careful test design it should be possible to calculate the values for all of these coefficients from the same set of data.

From the situation shown in Figure 3.9 the magnetic suspension torque can be represented as given in Equation (3.56).

51)

FLOAT

CASE

FLOAT CENTER OF MASS

FLOAT GEOMETRIC CENTER

CASE MAGNETIC SUSPENSION NULL POINT

CASE GEOMETRIC CENTER

(REFERENCE POINT)

FLOAT CENTER OF MAGNETIC SUSPENSION FORCES

FLOAT CENTER OF BUOYANCY

Fig.3.9 Representation of float and case reference points

MG = 2K,d,gs , (3.56)

where MQ is the torque (dyne-cm) about the output axis, K, is the stiffness (dyne/cm) of the magnetic suspension system along the input axis, d, is the displacement (cm) of the float from its reference point along the input axis and gs is the distance (cm) from the center of buoyancy to the center of magnetic suspension along the spin axis. Thus, two of the magnetic suspension torque coefficients on the torque generator end of the gyro can be represented as:

T, • 2K,gTS

(3.57) TS = 2KS8TI •

Similarly, the coefficients on the signal generator end of the gyro can be represented as:

S, = 2K,gss

SS = 2KS8SI • (3.58)

Although only these four error coefficients will be included in the following error analysis for simplicity, a more generalized expression can be obtained by a full expansion of Equation (3.55).

3.4.2.3 Analysis of compliance coefficient errors

The actual motion of the float with respect to the case during a tumble test will depend to a large extent on the temperature distribution within the floatation fluid. For this example, assume that the signal generator end of the gyro is slightly above floatation temperature and the torque generator end is slightly below floatation temperature. Assume also, that a two orientation torque-to-balance tumble test is performed with the output axis parallel to earth axis both north and south.

The generalized specific force inputs to the gyro due to gravity are29:

f| = —g cos L sin 0

fO = — g cos L sin A cos 0 4- g sin L cos A \ (3.59)

fs = —g cos L cos A cos 0 — g sin L sin A

51

where

L local astronomic latitude angle

A orientation of the gyro about its input axis starting with zero degrees at OA//EA North

0 rotation of the test table about minus earth axis starting with IA west at zero degrees.

For OA//EA north, the specific force inputs to the gyro are:

f| = —g cos L sin 0

fo = 4-gsin L

fs = —g cos L cos 0 .

For OA//EA south, the specific force inputs to the gyro are:

f| = —g cos L sin 0

fo = —g sin L

fs = + g c o s L cos 0 .

(3.60)

(3.61)

At the starting position for OA//EA north, the orientation of the float with respect to the case will appear as shown in Figure 3.10.

The locus of the float center with respect to the case center during a table axis rotation will be as shown in Figure 3.11.

The components of float displacement along the spin and input axes as a function of table angle are as follows:

1. OA//EA North

dSj = r s sin 0

(3.62)

2. OA//EA South

(3.63)

dSS " - r s c o s 0

d-p, = — r-j- sin 0

d j s = rT c o s 0 •

ds, = rs sin 0

dSS " - r s c o s *

d-pi = ~ t j sin 0

d-ps = r-p cos 0 .

The classical error model equation with the addition of the four magnetic suspension coefficients given in Equations (3.57) and (3.58) is written as:

Wd = W, + Dp + D,f, + D0f0 + Dsfs + D„f,2 + D 0 0 f 0 + D s sf s + DI0f,f0 + D,sf,fs +

+ Dosfefs + T|dTI + T s d T S + S,ds, 4- S s d s s . (3.64)

Substituting Equations (3.60) and (3.62), and Equations (3.61) and (3.63) in turn into Equation (3.64) gives the following expressions for the total gyro drift rate output, Wd .

For OA//EA North

w dn = Dp + j*(D*| + D s s )g 2cos 2L + D 0 g sin L + D 0 0 g 2 s i n 2 L + ( - D , g c o s L - jD I 0 g 2 s in 2L -

— T,rT + S*rs) sin 0 + (— D$g cos L — ^Dosg2-5**1 2L 4- T s r j — Ssrs) cos 0 4-

+ J*D,sg2cos2Lsin 20 + ±(DS S - D„)g2cos2L cos 20 . (3.65)

52

T . G . E N D S . G . END

Fig.3.10 Float orientation at start of OA//EA North

LOCUS

*>-lA

LOCUS

> - l A

T.G. END S.G, END

Fig.3.11 Locus of float center for OA//EA North

For OA//EA South

Wds = Dp 4- ^(D„ 4- Dss)g2cos2L - D0g sin L 4- D0 0g2sin2L 4- (-Dig cos L + iD I0g2sin 2L

— T,rT 4- Sjrg) sin 0 4- (Dgg cos L — ^Dosg2sin 2L + T§rj — Ssr$) cos 0 —

- i}DIsg2cos2 L sin 20 4- i ( D s s - Dn)g2cos2 L cos 20 .

The classical solution for the drift coefficients from these two orientations is obtained by assuming that DQO = 0 and D„ = —Dss • T*16 equations are as follows:

Dp =

(3.66)

D, -

Dn =

\ \

. AON

A

. AON

2

_ B1S

+ A0S

2

N + A i s 2 cos L

_ A0S

sin L

- B , N

(3.67) cont'd

2 cos L

53

DSS =

Dio =

Dis -

Dos =

B2N + 2 cos

A i s _

B2S 2L

A IN sin 2L

A2N _

cos2

BIN

A2S L

+ B,S

sin 2L

(3.67) cont'd

where A0 , A, , B, , A2 , and B2 are the Fourier coefficients for either the OA North (N) or the OA South (S) orientations.

If Equations (3.67) are used to solve for the drift coefficients and the magnetic suspension torque errors given in Equations (3.57) and (3.58) are present, the errors given in Equation (3.68) are introduced. The hat (") is used to represent the calculated coefficient while the value without the hat represents the actual drift coefficient.

D, = \ cos L

S,rT

)

^- •^ (Ma) . (3.68)

Using this particular example, the calculation of D Q S contains an error term which is proportional to the magnetic suspension torque terms. To obtain a feel for the amount of error that can be introduced into the calculation of D Q S , nominal values for the sensitivities and coordinate function magnitudes can be used. Assume that:

% = T$ = ±0.05 meru/uinch

rS = rT = ± 2 u inches .

Substituting these values into Equation (3.68) gives:

Dos = DQS * 0-2 meru .

(3.69)

(3.70)

This value is representative of the error to be expected during a two orientation tumble test. The use of other orientations and the consideration of other float position and velocity error terms will cause similar errors in the other classical error model coefficients26. For example, if SA//EA North and SA//EA South orientations are used, the magnetic suspension error torque calculated above for D Q S would appear in the D,s coefficient instead. Thus, it has been shown by this simple analysis that very significant errors can be introduced into the calculation of the compliance coefficients from tumble test data due to motion of the float. These errors can be reduced through mathematical modeling and compensation, or through a more rigid control of these error parameters during the gyro assembly process.

Although it will not be shown here, it is possible to measure the motion of the float during tumble testing using a selective set of orientations and to calculate the float position error coefficients simultaneously with the classical drift coefficients. The result of this calculation will be a substantial reduction of the residuals and a better description of the behavior of the gyro in all testing situations. A closer agreement with compliance coefficients obtained from linear vibration testing should also be possible.

3.4.3 New Concepts for Gyro Vibration Testing

3.4.3.1 Test description

Precision linear vibration tests, which are currently used to determine the gyro compliance coefficients are described in References 2 and 29. In these tests the gyro is oriented such that the input axis lies in the sterile plane and either the output or the spin axis is aligned with the earth rate vector. The vibratory accelerations are applied along an east-west axis using an electrodynamic shaker and slip table combination. The gyro is manually repositioned about the output axis (in the case of OA//EA) after each set of vibratory inputs are completed. Using eight positions (about earth axis) in each of three gyro orientations (about input axis), all six gyro compliance coefficients can be obtained with redundant estimates being obtained for D„ , D Q Q . ar*d Dss •

54

In each of the twenty-four positions defined above, vibratory inputs are applied to the gyro at several different acceleration levels at one fixed frequency. "Torque data is recorded for the same set of acceleration levels at each angular position. Angular position changes are made manually as are the settings of the acceleration levels at each position. After a complete cycle of angular positions, the data from that orientation is processed to extract the appropriate compliance coefficients2." The results obtained from this test yield a set of six compliance coefficients which are measured at one frequency, usually somewhere between 50 and 100 Hz.

Taking advantage of the suggestions made in Section 3.4.1.3 to use all of the test data, a gyro vibration test will be described in which the time varying gyro output signal is analyzed instead of using only the d.c. (rectified) portion. It will also be beneficial to use random vibration inputs instead of sinusoidal inputs. In this case all frequencies from d.c. to the upper cutoff frequency of the random noise generator will be applied to the test specimen during the test. The important advantage of using random noise is that the individual gyro drift coefficients as a function of frequency can be obtained. This means that it would be possible to obtain the transfer function of the gyro drift coefficients with respect to input accelerations. This would be the case for all of the drift coefficients, not only the compliances.

ORIENTATIONS: In order to design an efficient linear vibration test along the lines of that suggested above, a set of gyro positions must be determined which will provide all of the required data in an efficient manner. A convenient formulation for these positions is to follow the pattern described in the two-axis tumble test. Thus, for the gyro vibration test orientation pictured in Figure 3.12, the components of specific force along the gyro input, output and spin axes can be expressed as follows:

fj m cos 0 av 4- g cos L sin 0

fo = — sin 0 cos 7 av 4- g cos L cos 0 cos 7 — g sin L sin 7

fs = —sin 0 sin 7 av 4- g cos L cos 0 sin 7 + g sin L cos 7 .

(3.71)

If a suitable ratio of rotations about 0 and 7 is used, such as 5:2, 7:2, or 7:4, a solution for all of the drift coefficients could be obtained by the same procedure which was used in the two-axis tumble test and in the centrifuge test. Since all of the drift coefficients, except Dp , are to be obtained from the first and second harmonics of the vibrator input frequency, a constant rate which is sensed by the gyro will be of no consequence. Therefore, it would be possible to use a continuous two-axis tumble test during this vibration test. All of the gyro torque data should be bandpassed so that the low frequency components which tend to be corrupted by earth rate coupling, fixed restraint drift, gyro misalignment, and positioning device rate uncertainties are eliminated. The high-frequency cutoff of the bandpass filter could extend to the point where the slip table resonances become a problem or where data acquisition sampling frequency limitations are reached.

INSTRUMENTATION: A two-axis positioning fixture which is mounted on a vibration test fixture will not maintain a precise position and will most likely exhibit resonance characteristics of its own. It will be extremely important then, that the exact motion which is sensed by the gyro under test be accurately measured and recorded simultaneously with the gyro torque data. This can be done using a minimum array of six linear accelerometers or a combination of linear and angular sensors. By monitoring the six degrees-of-freedom of the random vibration motion which provides the excitation input to the gyro, it will be possible to simultaneously extract all of the drift coefficients of an expanded dynamic gyro error model through the use of cross-power spectral density analysis.

DATA COLLECTION: All of the test data including the motion sensor outputs along with the gyro torque data must be recorded on magnetic tape or directly to a disc file using a high-speed multiplexer and an analog-to-digital converter. The data should be collected over a reasonably long time period, such as five to ten minutes, to permit the two-axis positioning fixture to complete a fairly slow rotation cycle. Additional data will allow longer averaging times to be used and thus provide more stable estimates of the drift coefficients as will be described in the next section.

3.4.3.2 Data analysis

If the two-axis positioning fixture, which is mounted on the slip table, is rotated at a rate much less than one Hz, then the input rates which are sensed by the gyro will lie in this low frequency band. Consequently, the data which is contained in the frequency band between 5 and 100 Hz will be of most interest. Therefore, all of the recorded data channels should first be treated by a digital bandpass filter with this characteristic.

In order to describe how gyro drift coefficients (as a function of frequency) can be extracted from random input excitation, consider the gyro first as a single-input, single-output linear system. Assume that the system is subjected to a well-defined input x(t) . The output y(t) can be related to the input by Equation (3.72).

y(t) = J°° h (T)x ( t -T )d r . (3.72)

55

EARTH AXIS

NORTH

OUTPUT AXIS

EAST

Fig.3.12 Gyro orientation for linear vibration test

The auto-correlation function Rxx(v-) of the input x(t) is obtained from the relation:

Rxx(r) = J x ( t ) x ( t - r ) d r . (3.73)

The cross-correlation function Rxy(T) between the input x(t) and the output y(t + r) is obtained from the relation:

Rxy(r) = " x(t)h(.t)x(t + T-£)d.: . 'o

Substituting Equation (3.73) into (3.74) we obtain:

Rxy<» = J " h t t ) R x x ( r - j t ) d { .

(3.74)

(3.75)

The transformations of Equation (3.75) to a complex-valued frequency domain by taking the Fourier Transform yields the cross-spectrum relationship.

Sxy(f) = H(f)Sxx(f)-

In terms of a one-sided physically realizable spectral density function. Equation (3.76) becomes

Gxy(f) = H(f)Gxx(f).

From Equation (3.77) the system transfer function H(f) can be determined

Gvv(0

(3.76)

(3.77)

(3.78)

56

The frequency function H(f) is a complex quantity which can be represented as a magnitude and phase.

(C2(f) 4- Q2(f)l,/2

|H(f)l

4>(f) = tan

IGxx(OI

, Q(0 C(f)

(3.79)

where C(f) and Q(0 represent the co-spectrum estimate and the quad-spectrum estimate between x(t) and y(t)

Another useful quantity, the coherency spectrum, represents the correlation coefficient between the input and output variables as a function of frequency31.

72(0 = |Gx y( f) |

Gxx(f)Gyy(f)

The coherency function theoretically should satisfy 0 < 72(f) < 1 for all values of f .

(3.80)

Thus, if sufficient input excitation power is applied to the system at all frequencies of interest, then the input/ output relationship in terms of its magnitude and phase as a function of frequency can be established using Equation (3.79). This operation is relatively simple for a single-input, single-output linear system using a fast Fourier transform algorithm on a digital computer. The operation becomes rapidly more complex and time consuming as additional independent variables are added to the system representation.

Now assume in the case of the gyro performance model that nine correlated input variables and a single output variable representing gyro torque are measured during a random vibration test. The basic objective is to characterize the linear transformation properties between the nine inputs and the single output. An idealized schematic for this system is shown in Figure 3.13. To solve for the frequency response functions H,(f) through H9(f) , many complex valued quantities must be solved for. First, ten ordinary power spectral density functions must be computed, one for each input and the output. Then forty-five cross-spectral density functions must be computed. This includes thirty-six cross-spectral density functions between all the inputs, and nine between each of the inputs and the output. A 10x10 matrix of 100 elements is obtained with the lower-left elements being given as complex conjugates of the upper-right elements. This requires many computations, especially if many frequency values are used. References 32 and 33 provide some useful guidelines in the planning and computation of these power spectra.

The nine-dimensional input vector, X(t) , of the nine input time histories is represented as:

X(t) = [x,(t),x2(t), x,(t)] . (3.81)

The nine-dimensional cross-spectrum vector of the output y(t) with the inputs x,(t) is given as:

'xy = [G i y ,G y . ^ j y . *9J,1

The nine-dimensional frequency response function vector is:

H = [H,,H2, H9] .

Finally, the 9 x 9 matrix of the cross spectra of all the inputs x(t) is:

(3.82)

(3.83)

G „ G12

G2 1 G22

G 9 I G92

G.

(3.84)

From Equation (3.77), the formula that specifies the frequency response functions in terms of the cross-spectral density functions is;

(3.85) 'xy = GXXH'

The inversion of the complex matrix Gx x to obtain a solution for H , is quite an involved procedure, not be described here since most computing installations have their own subroutines available matrix is computed, the nine frequency response functions are obtained from the formula:

It will After this inverse

ll1 / - - i p 1 (3.86)

57

x 3(t)

x 9(t)

Nine

Inputs

y(t;

Single Output

Frequency

Response

Functions

Fig.3.13 Multiple-input, single-output gyro performance model relationship

The values obtained for these frequency response functions are valid whether or not the inputs are correlated. It is assumed however that the test has been designed such that the matrix G x x is nonsingular and the inverse exists. A more detailed description of this computational procedure can be found in References 34 and 35. Although continuous estimates of the drift coefficient frequency responses are possible if continuous rotations of the gyro are used this is not essential to obtain a single solution. The test data from several fixed positions can be combined to give the same results. This would certainly prove to be the easiest approach to use until suitable dynamic test fixtures can be designed and fabricated.

If the angular rate inputs to the gyro are large during the test, then an expanded performance model may be required to account for the rate sensitivity error terms. Such a model might be represented by Equation (3.87). Better dynamic models than this have been derived by considering the motion of the gyro float with respect to the case as independent forcing functions36. The use of an expanded model will, of course, increase the dimension of the matrix calculations.

Wd = co, 4- Dp + D,f, + D0f0 + Dsfs + D„f,2 4- D 0 0 f 0 4- D s sf s + DI0f,f0 4- Disf,fs + D o s f 0 f s +

+ RQCOQ + R S C J S + R„ w f 4- R O 0 C O Q 4- Rss^S + RISWIWS + R OS w O w S + A*W| 4- AQCOQ + A s w s . (3.87)

3.4.3.3 Summary

The above description of the data analysis procedures required to extract gyro drift coefficients from a random excitation linear vibration test is a radical departure from present test procedures. The extreme complexity of the data analysis task is amply rewarded by the amount of new information that is obtained concerning the test instrument. The availability of large high-speed digital computers make these computations fairly straightforward (although time-consuming) to perform. The advantages that can be achieved by using such a method over current techniques are:

(a) The gyro's fixed-drift, dynamic misalignment, and low-frequency error sources are relatively unimportant.

(b) Precise input vibration motion is not essential.

(c) The drift coefficients can be obtained in amplitude and phase over all frequencies that are excited.

(d) The gyro mass unbalances and the other odd-power specific force sensitive coefficients can be obtained.

(e) Considerably less time is required for the drift coefficient solutions.

(0 The short-term instabilities in the drift coefficients can be obtained if a continuous reorientation of the gyro is mechanized.

58

(g) The estimates of the drift coefficients can be improved considerably if multiple data records are used to obtain ensemble averages of the cross-power spectra.

(h) The sensitivity of the gyro to angular rates about its principal axes can be determined and accounted for in the data if an expanded error model is used,

(i) It may be possible to obtain near real-time coefficient calculations if a limited model is used and recursive estimation schemes can be realized.

Some of the additional requirements for the conduction of this type of vibration test are:

(a) The test specimen must be extensively instrumented to measure the total input motion.

(b) The angular rate about the gyro's input axis must be kept low.

(c) A high-speed data acquisition system is required.

(d) Sophisticated computer data reduction programs must be used.

(e) A suitable set of gyro test orientations must be devised so that the correlation between the error model coordinate functions is minimized.

3.4.4 Conclusions

The intent of this section on the linear vibration testing of gyros was to point out the need for new and improved test methods. It has been shown that other test arrangements and data analysis procedures are possible for linear vibration. The tendency here has been to stress the use of more analytical sophistication and to de-emphasize the use of more expensive test equipment to achieve stringent mechanical accuracies. The increased use of monitoring and model fitting will help to achieve additional insight into the underlying problems causing discrepancies to appear between various gyro testing methods. The evaluation of the gyro mass unbalance coefficients at levels greater than one-g on the linear vibrator should prove to be extremely beneficial.

Several suggestions have been made which should be considered in the development of future linear vibration test facilities. These include (i) the use of opposing shakers, (ii) continuous reorientation of the gyro during vibration, (iii) fully implemented digital control of the shaker, input spectrum, data acquisition and processing, (iv) random noise excitation, (v) installation of a complete set of motion sensors, and (vi) system level testing of components. The measurement of the drift coefficient frequency response functions would certainly add a considerable amount of new information to the existing body of linear vibration data. The investigation of compliance coefficient nonlinearity with input amplitude and the behavior of higher order coefficients can also be considered.

4. CONCLUSIONS

4.1 Summary

This paper has presented some recent thinking on the testing of advanced precision inertial gyroscopes. The basic test considerations dealing with the environment, excitation, monitor and evaluation have been discussed. Considerable attention has been given in Chapter 2 to techniques which have been employed in solving the base motion isolation and control requirements and in describing a new software concept for automating developmental gyro tests. Chapter 3 describes several new testing techniques for gyros, including the two-axis tumble test, centrifuge test and linear vibration test. A description of the test procedures and an outline of the appropriate analytical methods to be used in the reduction of test data are given for each test. It appears that all of these new test methods will require the use of an expanded gyro error model to eliminate the misleading effects of interfering inputs.

With the introduction of an advanced series of inertial sensors, component testing is entering a new phase of its evolutionary process. Testing concepts which were devised during a time period when solid state electronics, precision measurement transducers, versatile data acquisition systems, and inexpensive yet powerful laboratory computers were not available, are being replaced by modern techniques which take full advantage of recent equipment and analytical advancements. The whole problem of inertial component testing is being reviewed in the light of these new considerations. A fundamental understanding of the real problems to be faced in the evaluation of these ultra-precise inertial devices must be obtained in order that the most desirable new approaches can be formulated. Until the nature and source of the error generating mechanisms which are in present instruments are thoroughly investigated, it will not be possible to produce more accurate inertial devices.

Expanded error models are being used to describe the basic physical relationships which are interacting within an instrument to generate the erroneous results which are observed in the output signal. The real limit of an instrument's performance can be ascertained only after all significant error sources have been modeled and compensated. This information can then be used by the instrument designer to make a more meaningful trade-off between the various design parameters.

59

4.2 Future Trends

The requirement for the testing of precision inertial sensors will continue for many more years. Therefore, I would like to offer some additional thoughts on how I envision future advancements will take place.

Most evident is the fact that a considerable increase in the use of on-line digital computers for control, processing and display will occur. Almost no aspect of the testing process will avoid its eventual takeover. This includes the monitor and control of such things as room temperature, humidity, base motion, input power, test sequencing, data acquisition, processing and display. We are only scratching the surface of what is to come in computer control and software sophistication. This means that a larger percentage of the initial effort will be spent in software development and testing. The test engineer however, will be able to implement new test procedures almost as soon as they are conceived and he can observe the results of these tests almost as soon as the test is completed. This will give him much more time to concentrate on the real meaning of what has been measured, rather than consuming much of his time in routine data manipulation and handling.

A greater awareness of base motion effects is already prevalent at most test installations. New seismic sensors are being developed to measure the most disturbing aspects of this environmental factor. These devices will eventually be integrated as a standard part of the supporting test hardware. The use of an auxiliary gyro to measure and extract base motion errors from the output of the test specimen does not appear to provide a long range solution for the highest quality sensors. Specialized devices such as seismometers, tiltmeters, angular and linear accelerometers, and velocimeters can be used to separate the disturbing motion into several frequency regions and input directions. Without all of the size, weight, reliability and extreme environmental requirements that are placed on operational gyros, these specialized devices can be made at much lower cost and with superior performance characteristics for the intended purpose.

More emphasis is being placed on expanding the present gyro error model. More variables will be instrumented and investigated than ever before. New methodologies and analytical tools will be employed to aid in this modeling process. The important and consistently observed error sources will gradually find their way into the accepted model. These improved models however, must be verified in many different test environments with consistent results before they can be universally accepted.

The important questions concerning the effects of base motion vibrations on gyro calibrations and alignment will be pursued. Dynamic models may be developed with the aid of the special motion sensors described above to obtain a more exact understanding of this problem area. Optimum orientations and data processing procedures could be developed to minimize these errors in a particular seismic environment.

Along this same line, a trend toward optimal test design will minimize the amount of data which must be taken in order to satisfy a given set of test objective criteria. Considerably less testing time will be required to obtain solutions for the individual error model coefficients. Real-time digital computer control of the test sequencing, data analysis and display will be used to mechanize these schemes.

For many types of gyro tests, component testing will slowly give way to system level testing. This is mainly because of the protection and control of the test environment which is provided by the system. The centrifuge and vibration testing areas, where uncontrollable dynamic motion about all gyro axes is most prominent, are prime candidates for this change. The results which can be obtained from the system level testing of components should be (i) more representative of the actual operational performance of the instrument, (ii) less expensive to perform, and (iii) more meaningful and reliable.

This also leads to the fact that there will be greater emphasis on high-g testing using new equipment and new analytical procedures. Modeling and compensation will be indispensable to the successful conduction of these quantitative tests. The conventional g sensitive and g-squared sensitive terms as a function of higher g levels over a broad frequency band will attract more attention. Test methods which yield gyro error model coefficients over much shorter time intervals should become practical using this high-g test equipment.

4.3 Recommendations

Considering the direction that precision gyro testing will take in the future, several recommendations are given here for improving our present test capability. The developmental items that require long lead-times, should be initiated first. This refers to areas where improvements are slow to make and even a large amount of money does little to speed up the learning process. Even if conducted on a low key effort, extremely important investigations should be continued into the best methods for controlling room ambient temperature, pressure, acoustics, magnetic fields, base motion, electrical grounding systems and RF interference. Electronic sensors and modem servo technology should be applied to these long-standing process control problems. Control of the immediate test table temperature environment should be investigated using improved insulation material, temperature sensors and control devices. Control of time varying magnetic fields at critical locations could be servoed out using an electronic Gaussmeter and Helmholtz coils. Different test facilities may be able to achieve better results than others in certain of these areas due to their particular type of construction. Therefore a liberal cross-exchange of information should

60

take place until some accepted practices can be established for regulating the numerous environmental disturbance factors. Facilities should then be improved or rebuilt according to these new guidelines.

Computer software will become a large test capability investment in the future. In addition, it takes considerable time to develop and thoroughly evaluate. Consequently, the computer hardware should be selected with great care. It should (i) possess a maximum reserve capacity to meet continually more demanding requirements, (ii) be designed with an expansion capability, (iii) be equipped with a generalized and versatile software operating system, (iv) have the highest possible continuous duty reliability, and (v) be relatively easy to interface and maintain.

The base motion environment should be carefully studied; improved seismic sensors should be evaluated: and a complete instrumentation system should be developed. Major test platform modification should be gradually accomplished.

Current test electronics hardware should be thoroughly scrutinized in terms of accuracy, versatility and reliability. The excitation, control and monitor functions should be made programmable as much as possible.

Additional studies into the continuous measurement of low-frequency inertial azimuth variations should be made. The combination of a vertical gyroscope and a well-modeled, four-position gyro compassing system should be investigated as a possible long-range solution to this problem.

Perhaps our basic thinking process in the evaluation of inertial sensors should be altered. Novel test techniques and analytical methods will probably be required if the testing challenge which is presented by future precision inertial sensors is to be met.

REFERENCES

1. Draper, C.S. Background for Specification. Engineering and Operational Realization of Inertial Sensors to Meet the Requirement of High Quality Control. Navigation and Guidance Systems Adaptable to Marine. Aeronautical and Astronautical Vehicles of all Kinds. MIT Draper Laboratory Report R-623, September 1968.

2. Denhard, W.G. (Editor)

Inertial Component Testing: Philosophy and Methods. AGARDograph No.l 28, Technivision Services, Slough, England, 1970.

3. Gray, et al.

Earth Motions and Their Effects on Inertial Instrument Performance. AFCRL TR 72-0278, Bedford, Massachusetts, 27th April 1972.

4. Riley, C.E. Vibration Survey of Air-Supported Isolation Platforms. AIAA Paper No.68-893, AIAA Guidance, Control, and Flight Dynamics Conference, Pasadena, California, August 1968.

5. Crowley, et al.

Precursory Siloed Missile Geokinetic Study, Hill Air Force Base. Utah. AFCRL TR 72-0531, Bedford, Massachusetts, 5th September 1972.

6. Alsup, Wilson

Ground Disturbance from Heavy Vehicles and Well Drilling Activities. TR No.64-48, The Geotechnical Corporation, Garland, Texas, April 1964.

7. Weinstock. H. Limitations on Inertial Sensor Testing Produced by Test Platform Vibrations. NASA Technical Note TN D-3683, Cambridge, Massachusetts. November 1966.

8. Morrison, R. Grounding and Shielding Techniques in Instrumentation. John Wiley and Sons Inc., New York, 1967.

9. Crowley, et al.

10. Blandford, Ruskey

1 I. Preskitt, S., Fix, J.

12. DeBra, D.

An Analysis of the Vibration Environment at Northrop's Norwood, Massachusetts Test Facility with Application to Gyro Testing. AFCRL Report 70-0355, June 1970.

Experimental Determination of Test Pad Performance. AIAA Paper No.68-884, AIAA Guidance, Control, and Flight Dynamics Conference, Pasadena, California, August 1968.

Six-Degree-of-Freedom Test Podium at the United States Air Force Standards Calibration Laboratory. Paper presented at the Geokinetics Subcommittee Meeting, August 1963.

A Precision, Active Table Leveling System. Paper presented at the AIAA Guidance and Control Specialist Conference, August 1966.

(.1

13. Pepi, J., Cavanaugh, R.

14. Tsutsumi, K.

Performance Characteristics of an Automatic Platform Tilt Stabilization and Vibration Isolation System. AIAA Paper No.67-548, August 1967.

A Ground Tilt Isolation Platform. MIT Draper Laboratory Report No.E-1 508, January 1964.

15. Weinstock, H.

16. Wittry. J.P.

17. Lorenzini, D.A. Neeland. R.

18. Lorenzini, D.A. Schunk

19. Russell. J.F.

20. Koestler, J.G.

Design of a Precision Till and Rotational Vibration Isolation System for Inertial Sensor Testing. AIAA Paper No.68-894, August 1968.

Description of an Inertial Test Facility Incorporating a Passively Isolated and Actively Stabilized Platform. AIAA Paper No.69-863, August 1969.

Analysis of a Pneumatic Isolation System for Inertial Instrument Testing. AIAA Paper No.71-910, August 1971.

Seismometer Compensation for Acceleration Measurements. FJSRL TR-72-0015, USAF Academy, Colorado, August 1972.

Gyroscope Standard Torque-to-Balance Test. Report MDC-TR-67-79, Holloman AFB, New Mexico, June 1967.

A New Method for Determining Short-Term Drift Characteristics Fourth Inertial Guidance Test Symposium Proceedings, Holloman AFB, New Mexico, 6-8th November 1968.

21. Draper, C.S., et al.

23. Wilkinson. R.H.

24. Wilkinson. R.H.

25. Kaiser, K.W.

26. Lorenzini, D.A.

27. Wrigley, et al.

28. Wimber, B.J.

29. Palmer, P.J.

30. Crawford, B.S.

31. Bendot, Piersol, A.G.

32. Blackman, Tukey

33. Piersol, A.G.

Torque Induced by Convective Effects Acting on a Cylinder Floated in a Thin Layer of Viscous Fluid. Technical Report AFAL-TR-68-171, Wright Patterson AFB, Ohio. August 1968.

Hydrodynamic Force Considerations of Coaxial and Nearly Coaxial Cylinders of Finite Length. GR 13, Gyro Research Group, MIT/DL, August 1951.

Some Sources of Error in the Testing of Floated Single-Degree-of-Freedom Gyroscopes. GL-389, MIT/DL, or 4th Biannual Guidance Test Symposium, Holloman AFB. New Mexico, April 1968.

Study of Surface Thermal Gradients on Instrument Structural Elements. MIT/DL Report T-416, Cambridge, Massachusetts, May 1965.

Optimal Control of Steady-State Surface Temperature Distribution. MIT/DL Report T-500, Cambridge. Massachusetts, April 1968.

Gyro Error Modeling. MIT/DL Report T-530, Cambridge, Massachusetts, March 1970.

Gyroscopic Theory. Design and Instrumentation. MIT Press, Cambridge, Massachusetts, 1969.

The Apparent Input Axis Misalignment Error Caused by Angular Rotation About the Output Axis of a Single-Degree-of-Freedom. Rate-Integrating Gyro Proceedings of the Fourth Incrtial Guidance Test Symposium, Holloman AFB, New Mexico, November 1968.

Gyro Torque Coefficients. MIT Draper Laboratory Report E-1601, Cambridge, Massachusetts, January 1966.

Analysis of Strapdown Sensor Testing. TASC TR-147-2, The Analytic Sciences Corp., Reading, Massachusetts, June 1970.

Measurement and Analysis of Random Dala. John Wiley and Sons Inc., New York, 1966.

The Measurement of Power Spectra. Dover Publications Inc., New York, 1958.

The Measurement and Interpretation of Ordinary Power Spectra for Vibration Problems. NASA CR-90, Measurement Analysis Corporation, Los Angeles, California, September 1964.

<>:

34. Enochson, L.D.

35. Goodman, N.R.

36.

Frequency Response Function and Coherence Functions for Multiple Input Linear Systems. NASA CR-32, Washington, D.C., April 1964.

Measurement of Matrix Frequency Response Functions and Multiple Coherence Functions. TR-AFFDL-TR-65-56, Measurement Analysis Corporation, Los Angeles, California, June 1965.

Dynamic Errors in Strapdown Inertial Navigation Systems. NASA-CR-1962, The Analytical Sciences Corporation, Reading, Massachusetts, December 1971.

AGARDograph No. 192 Advisory Group for Aerospace Research and Development, NATO TESTING OF PRECISION INERTIAL GYRO­SCOPES Dino A. Lorenzini Published June 1974 72 pages, including figures

Testing techniques are of particular importance to the field of precision inertial gyroscopes and this text book covers the subject in two principal parts. The first examines Basic Test Considerations, i.e. Environment, Excitation, Monitor and Evaluation. The second part deals with the general area of Gyro

P.T.O.

AGARDograph No. 192 Advisory Group for Aerospace Research and Development, NATO TESTING OF PRECISION INERTIAL GYRO­SCOPES Dino A. Lorenzini Published June 1974 72 pages, including figures

Testing techniques are of particular importance to the field of precision inertial gyroscopes and this text book covers the subject in two principal parts. The first examines Basic Test Considerations, i.e. Environment, Excitation, Monitor and Evaluation. The second part deals with the general area of Gyro

P.T.O.

AGARD-AG-192 681.082.16:621.001.4

Gyroscopes Inertia Evaluation Tests Test facilities Vibration tests

AGARD-AG-192 681.082.16:621.001.4

Gyroscopes Inertia Evaluation Tests Test facilities Vibration tests

AGARDograph No. 192 Advisory Group for Aerospace Research and Development, NATO TESTING OF PRECISION INERTIAL GYRO­SCOPES Dino A.Lorenzini Published June 1974 72 pages, including figures

Testing techniques are of particular importance to the field of precision inertial gyroscopes and this text book covers the subject in two principal parts. The first examines Basic Test Considerations, i.e. Environment, Excitation, Monitor and Evaluation. The second part deals with the general area of Gyro

P.T.O.

AGARDograph No. 192 Advisory Group for Aerospace Research and Development, NATO TESTING OF PRECISION INERTIAL GYRO­SCOPES Dino A.Lorenzini Published June 1974 72 pages, including figures

Testing techniques are of particular importance to the field of precision inertial gyroscopes and this text book covers the subject in two principal parts. The first examines Basic Test Considerations, i.e. Environment, Excitation, Monitor and Evaluation. The second part deals with the general area of Gyro

P.T.O.

AGARD-AG-192 681.082.16:621.001.4

Gyroscopes Inertia Evaluation Tests Test facilities Vibration tests

AGARD-AG-192 681.082.16:621.001.4

Gyroscopes Inertia Evaluation Tests Test facilities Vibration tests

Testing Techniques and provides a sample of current thinking in the areas of tumble testing, error modeling, centrifuge and linear vibration testing.

This AGARDograph was prepared at the request of the Guidance and Control Panel of AGARD-NATO, in order to disseminate the content of a Short Course presented, in April 1973, in Italy. Netherlands, Germany, Belgium, United Kingdom and Norway.

Testing Techniques and provides a sample of current thinking in the areas of tumble testing, error modeling, centrifuge and linear vibration testing.

This AGARDograph was prepared at the request of the Guidance and Control Panel of AGARD-NATO, in order to disseminate the content of a Short Course presented, in April 1973, in Italy, Netherlands, Germany, Belgium, United Kingdom and Norway.

Testing Techniques and provides a sample of current thinking in the areas of tumble testing, error modeling, centrifuge and linear vibration testing.

This AGARDograph was prepared at the request of the Guidance and Control Panel of AGARD-NATO, in order to disseminate the content of a Short Course presented, in April 1973, in Italy, Netherlands, Germany, Belgium, United Kingdom and Norway.

Testing Techniques and provides a sample of current thinking in the areas of tumble testing, error modeling, centrifuge and linear vibration testing.

This AGARDograph was prepared at the request of the Guidance and Control Panel of AGARD-NATO, in order to disseminate the content of a Short Course presented, in April 1973, in Italy, Netherlands, Germany, Belgium, United Kingdom and Norway.

AGARDograph No. 192 Advisory Group for Aerospace Research and Development, NATO TESTING OF PRECISION INERTIAL GYRO- f

SCOPES Dino A.Lorenzini Published June 1974 72 pages, including figures

Testing techniques are of particular importance to the field of precision inertial gyroscopes and this text book covers the subject in two principal parts. The first examines Basic Test Considerations, i.e. Environment, Excitation, Monitor and Evaluation. The second part deals with the general area of Gyro

P.T.O.

AGARDograph No. 192 Advisory Group for Aerospace Research and Development, NATO TESTING OF PRECISION INERTIAL GYRO­SCOPES Dino A.Lorenzini Published June 1974 72 pages, including figures

Testing techniques are of particular importance to the field of precision inertial gyroscopes and this text book covers the subject in two principal parts. The first examines Basic Test Considerations, i.e. Environment, Excitation, Monitor and Evaluation. The second part deals with the general area of Gyro

P.T.O.

AGARD-AG-192 681.082.16:621.001.4

Gyroscopes Inertia Evaluation Tests Test facilities Vibration tests

AGARD-AG-192 681.082.16:621.001.4

Gyroscopes Inertia Evaluation Tests Test facilities Vibration tests

AGARDograph No. 192 Advisory Group for Aerospace Research and Development, NATO TESTING OF PRECISION INERTIAL GYRO­SCOPES Dino A.Lorenzini Published June 1974 72 pages, including figures

Testing techniques are of particular importance to the field of precision inertial gyroscopes and this text book covers the subject in two principal parts. The first examines Basic Test Considerations, i.e. Environment, Excitation, Monitor and Evaluation. The second part deals with the general area of Gyro

P.T.O.

AGARDograph No. 192 Advisory Group for Aerospace Research and Development, NATO TESTING OF PRECISION INERTIAL GYRO­SCOPES Dino A.Lorenzini Published June 1974 72 pages, including figures

Testing techniques are of particular importance to the field of precision inertial gyroscopes and this text book covers the subject in two principal parts. The first examines Basic Test Considerations, i.e. Environment, Excitation, Monitor and Evaluation. The second part deals with the general area of Gyro

P.T.O.

AGARD-AG-192 681.082.16:621.001.4

Gyroscopes Inertia Evaluation Tests Test facilities Vibration tests

AGARD-AG-192 681.082.16:621.001.4

Gyroscopes Inertia Evaluation Tests Test facilities Vibration tests

Testing Techniques and provides a sample of current thinking in the areas of tumble testing, error modeling, centrifuge and linear vibration testing.

This AGARDograph was prepared at the request of the Guidance and Control Panel of AGARD-NATO, in order to disseminate the content of a Short Course presented, in April 1973, in Italy. Netherlands, Germany, Belgium, United Kingdom and Norway.

Testing Techniques and provides a sample of current thinking in the areas of tumble testing, error modeling, centrifuge and linear vibration testing.

This AGARDograph was prepared at the request of the Guidance and Control Panel of AGARD-NATO, in order to disseminate the content of a Short Course presented, in April 1973, in Italy, Netherlands, Germany, Belgium, United Kingdom and Norway.

Testing Techniques and provides a sample of current thinking in the areas of tumble testing, error modeling, centrifuge and linear vibration testing.

This AGARDograph was prepared at the request of the Guidance and Control Panel of AGARD-NATO, in order to disseminate the content of a Short Course presented, in April 1973, in Italy, Netherlands, Germany, Belgium, United Kingdom and Norway.

Testing Techniques and provides a sample of current thinking in the areas of tumble testing, error modeling, centrifuge and linear vibration testing.

This AGARDograph was prepared at the request of the Guidance and Control Panel of AGARD-NATO, in order to disseminate the content of a Short Course presented, in April 1973, in Italy, Netherlands, Germany, Belgium, United Kingdom and Norway.

DISTRIBUTION OF UNCLASSIFIED AGARD PUBLICATIONS

NOTE: Initial distributions of AGARD unclassified publications are made to NATO Member Nations through the following National Distribution Centres. Further copies are sometimes available from these Centres, but if not may be purchased in Microfiche

or photocopy form from the Purchase Agencies listed below. THE UNITED STATES NATIONAL DISTRIBUTION CENTRE (NASA) DOES NOT HOLD STOCKS OF AGARD PUBLICATIONS, AND APPLICATIONS FOR

FURTHER COPIES SHOULD BE MADE DIRECT TO THE APPROPRIATE PURCHASE AGENCY (NTIS).

NATIONAL DISTRIBUTION CENTRES

BELGIUM Coordonnateur AGARD - VSL Etat-Major de la Force Aerienne Caserne Prince Baudouin Place Dailly, 1030 Bruxelles

CANADA Defence Scientific Information Service Department of National Defence Ottawa, Ontario K1A OZ3

DENMARK Danish Defence Research Board Osterbrogades Kaseme Copenhagen O

FRANCE O.N.E.R.A. (Direction) 29, Avenue de la Division Leclerc 92, Chatillon sous Bagneux

GERMANY Zentralstelle fur Luftfahrtdokumentation und Information 8 Munchen 86 Postfach 860881

GREECE Hellenic Armed Forces Command D Branch, Athens

ICELAND Director of Aviation c/o Flugrad . Reykjavik

ITALY Aeronautica Militare Ufficio del Delegate Nazionale all'AGARD 3, Piazzale Adenauer Roma/EUR

LUXEMBOURG See Belgium

NETHERLANDS Netherlands Delegation to AGARD National Aerospace Laboratory, NLR P.O. Box 126 Delft

NORWAY Norwegian Defence Research Establishment Main Library P.O. Box 25 N-2007 Kjeller

PORTUGAL Direccao do Service de Material da Forca Aerea Rua de Escola Politecnica 42 Lisboa Attn: AGARD National Delegate

TURKEY Turkish General Staff (ARGE) Ankara

UNITED KINGDOM Defence Research Information Centre Station Square House St. Mary Cray Orpington, Kent BRS 3RE

Microfiche or Photocopy

National Technical Information Service (NTIS) 5285 Port Royal Road Springfield Virginia 22151, USA

UNITED STATES National Aeronautics and Space Administration (NASA) Langley Field, Virginia 23365 Attn: Report Distribution and Storage Unit (See Note above/

PURCHASE AGENCIES

Microfiche

ESRO/ELDO Space Documentation Service European Space Research Organization 114, Avenue Charles de Gaulle 92200 Neuilly sur Seine, France

Microfiche

Technology Reports Centre (DTI) Station Square House St. Mary Cray Orpington, Kent BRS 3RE England

Requests for microfiche or photocopies of AGARD documents should include the AGARD serial number, title, author or editor, and publication date. Requests to NTIS should include the NASA accession report number.

Full bibliographical references and abstracts of AGARD publications are given in the following bi-monthly abstract journals:

Scientific and Technical Aerospace Reports (STAR), published by NASA, Scientific and Technical Information Facility P.O. Box 33, College Park Maryland 20740, USA

Government Reports Announcements (GRA), published by the National Technical Information Services, Springfield Virginia 22151, USA

* Printed by Technical Editing and Reproduction Ltd Harford House. 7-9 Charlotte St, London. W1P 1HD